In the second episode of Yes! Precure 5, when Rin is asked to become the second Pretty Cure, her response is "No thanks, I'll pass." Eventually, she is forced into it when Nozomi is in danger and Rin is the only one around who could save her. Karen, too, refuses to try again when her pinky catch breaks apart.
It happens in Heart Catch Precure in the same exact way, except Erika didn't want to because of other responsibilities. She changes her mind when she sees how awesome it is (and when she sees Tsubomi get herself into trouble.)
Itsuki's the same way - Potpourri chose her to be a Precure, but she didn't feel she was worthy. It isn't until her brother is turned into a monster that she takes it up.
Then, there's Yuri, the former Cure Moonlight. The first time, Tsubomi's grandma has to remind Tsubomi and Erika that she can't transform. The second time, Potpourri tries to recruit her, but she turns it down. It isn't until the Precure Trials that she's dragged back, kicking and screaming.
Alice of DokiDoki! Precure refuses to become Cure Rosetta initially due to the fact that she kinda has rage control problems, stemming from the fact that she beat down a few bullies when she was younger.
Kouji Kabuto Jumped at the Call in the original Mazinger Z. In the first timeline seen in Shin Mazinger Zero, he refused the call and we got to see the outcome. It was not pretty (unless your definition of pretty includes the planet scorched away and the humanity wiped out by Mazinger-Z).
In AIR, once Yukito realizes that Misuzu's illness is killing her because she's become his friend, he tries to leave and never see her again, but then remembers that he has to try to save her. Also, in the movie, he did not even want to look for the girl in the sky.
Zakuro in Tokyo Mew Mew. She was already a Mew Mew by this point — her "refusal" concerned joining the team.
Usagi/Serena in Sailor Moonjust wanted to be normal, only agreed to get powers in order to save her friend, and even got out of having them once when she died at the end of the original anime's first series. She never got out of it in the manga, though she killed herself after being forced to kill her lover. They were both reborn and she had to fight again immediately afterward.
Haruka (Uranus) in the first anime refuses until a daimon is about to kill Neptune. She is warned not to say yes just to save them because she can't take it back once she does. She says yes finally anyway.
Ami (Mercury) in Pretty Guardian Sailor Moon refuses because she believes Usagi only befriended her so that she'd join up. As usual, a battle forces her to make the choice - although afterward, she claims to Usagi that she really does want to fight with her, probably still hoping it will let them become friends.
Nakajima Youko in The Twelve Kingdoms spends an unusually long portion of the entire first Story Arc of the series refusing the call in various ways.
Ginrai in Transformers Super God Masterforce initially refuses to get involved in the Autobot/Decepticon war, and only changes his tune when Hydra and Buster kill some of his friends while trying to get to him.
Asuna of Mahou Sensei Negima! seems to fit this. When she first finds out about magic, she wants nothing to do with it, and seems downright offended by how it's going against her nice and orderly world (the fact that it seems to be losing her crush isn't helping.) She actually falls into the "did this before, but forgot about it" category, having erased her memory ten years prior. Needless to say, given that the principal put Negi (The Call personified as a cute boy) in her room and he won't hang up, she gives in eventually.
Chisame puts up even more of a fight, and sort of succeeds in getting away from the weirdness...only to realize that now her life is incredibly boring, so she heads back in.
The Lucifer and Biscuit Hammer: Yuuhi gets picked to be the Lizard Knight, protect the princess, save the world, and so forth but flat out refuses to have anything to do with the whole affair and throws his spiritual advisor out the window (several times). That doesn't stop golems from trying to kill him, though, and the Princess's oath to destroy the world herself finally convinces him to participate wholeheartedly.
Kamina Ayato from RahXephon waffles between this and being willing to fight throughout the show.
In Prétear, Himeno at first doesn't even believe the Leafe Knights when they tell her she's the titular Magical Girl, thinking they're pulling a prank. When a monster actually shows up, she's rebuffed by the Tall, Dark and Snarky leader of the knights and runs off—but then returns when she decides she actually does want to help.
One side hero in Fist of the North Star, name of Jyuza, adhered to his trope. Jyuza is part of the Nanto Five Chariot Stars who defend the Last Nanto General. When Jyuza is sent a note telling him that he must jump into action to stop Raoh from getting to the Last Nanto General....he slices up the note saying "how stupid!", claiming that he'd rather be as free as a cloud. Course, since this is the series where Anyone Can Die, he's gotta answer the call sooner or later. You know what that means...
In Fushigi Yuugi, Chiriko intially ignores the mark on his foot designating him as a Suzaku warrior because he's terrified of what duties will be placed on him. Quite a natural reaction, considering he's ten years old; in fact, this is his default state whenever the mark isn't glowing. Which makes it all the more horrific when he's forced to kill himself as a Taking You with Me sacrifice later in the series.
This was a serious matter during the period of Fushigi Yuugi Genbu Kaiden, where a superstition was placed on the Priestess of Genbu and her Genbu Celestial Warriors. Indicating them as a curse, and by having them around will destroy the country. So whenever the priestess Takiko found one warrior and tries to persuade him to join, that warrior would automatically refuse the calling, simply because being awakened as a Celestial Warrior has already put their own lives into jeopardy for as long as they can remember.
In the first arc of 20th Century Boys, Kenji tries to put the band back to together, with partial results. A few join out of curiosity or loyalty, but at least one initially back out out of duty for family and jobs. Of course, they all join him later on after the timeskips
In Neon Genesis Evangelion, Shinji at first refuses to pilot Unit 01, out of a mixture of fear, confusion, and anger over the circumstances that got him to that point, only to have his father essentially guilt him into the task by forcing a very badly injured Rei to try to crawl out of her gurney to take his place. He ultimately agrees to pilot.
He also quits later after his father crosses the Moral Event Horizon during the Bardiel incident only to return as Zeruel curb-stomps the rest of NERV.
Katekyo Hitman Reborn! is basically an endless cycle of Tsuna refusing the call, his friends being put in danger, and him thus being forced to accept the call for just long enough to deal with this new threat. Each arc starts off this way, and several individual battles are sparked by near-identical internal struggles.
In Puella Magi Madoka Magica, Madoka, the title character of the show, is understandably resistant in accepting a contract to become a Magical Girl. Which is kind of the point of the series, as it becomes increasingly clear as the series goes on that being a Magical Girl is not a good thing.
In Silent Möbius (the TV series and films, not so much the Manga) Katsumi Liqueur not only refuses the call, but once she's been forced into answering the call, she makes a break for it with disturbing regularity.
Kyouya in Demon City Shinjuku is initially uninterested when Old Master Agni Lai asks him to face his father's Rival Turned Evil and save the world from being overrun by demons. Master Lai doesn't exactly help his case by predicting that there's no way Kyouya can win at his current level of skill, which irks Kyouya enough that he refuses the call a second time when it's delivered by Sayaka Rama, whose father was attacked by the Big Bad. Only when Sayaka ventures into danger herself does Kyouya relent and go after her to take up the mission.
THE iDOLM@STER - Ritsuko really doesn't want to replace Azusa on the Ryuugu Komachi unit when she gets the mumps. Thing is, she doesn't have a choice.
In Tenchi in Tokyo, Tenchi's the latest line of protectors of the Earth (instead of the long-lost prince of Jurai that he is in other canons focusing on him) and is given aTransformation Trinket in the form of seven jewels that turn into a BFS. Instead, he'd rather be a normal teenager and gives away one of the jewels to the six alien girls. This turns out to be crucial in Big Bad Yuugi's plans.
In Oregairu, HachimanHikigaya is adverse to joining the service club and leaving his self-imposed solitude. Good or bad, the call can make sure he doesn't graduate in only 3 years.
Bleach: Ryuuken Ishida is a quincy. Not just any quincy, but one powerful enough to curbstomp his very gifted son, Uryuu. However, he utterly refuses to use his power and the only time he was ever asked why, his response consisted of Blatant Lies that left Uryuu angry with him for years. Uryuu's grandfather did once say Ryuuken's refusal to be a quincy resides with either some kind of Declaration of Protection or an Obstructive Code of Conduct but what has yet to be explained. With the quincies and shinigami currently at war and Ryuuken apparently wanting nothing to do with either side, the story has made him a clear case of Heroic Neutral and a very strong contender for a Neutral No Longer fate.
Ultimate Spider-Man has Peter Parker mimicking his mainstream counterpart and the results thereof. The second Spider-Man, Miles Morales, refused to do anything with his powers; when Peter died, Miles figured that if he'd embraced his powers earlier, he might have brought aid to New York's hero, since their powers were connected. He decides to make up for it by filling the gap Peter left.
Watchmen. A major character arc for Silk Spectre and Nite Owl. Laurie Jupiter (Spectre) always felt pushed into being a crime-fighting heroine by her mother and is glad for an excuse to give up the role when the Keene Act is passed. Dan Dreiburg (Nite Owl) is less happy, but accepts that the days of masked vigilantes are over. Over the course of events however they remember just how good it was to fight bad guys and rescue people, and by the end have joined up to become a Battle Couple.
Sunfire was originally part of the famed All-New, All-Different X-Men, but quit the team after just one issue. A major part of the current Uncanny Avengers series involves Wolverine getting Sunfire to join The Avengers in order to finally live up to his potential.
ThisNodwick strip really says it all. Yeagar is prone to this sort of behaviour if there's no profit (or attractive ladies) involved.
The Authority. In the Jenny Sparks: The Secret History of The Authority mini-series, Jeroen Thornedike's refusal to be the next Doctor is why all the past Doctors think he's perfect for the job. It eventually culminates in Jenny Sparks having to shoot herself in the temple and be revived just for him to answer the Call.
In issue 50 of W.I.T.C.H., the Oracle shows the Guardians an Alternate Universe where Will rejected the Heart of Kandrakar when Yan Lin offered it (something that really freaked out the girls). Inverted later on in that world as that Heart is very persistent...
Also, Cornelia refused at first, believing it a prank from Yan Lin and only accepting after accidentally using her powers (proving it wasn't a prank) and realizing that her best friend Elyon was in trouble. Later in the series Cornelia is the only one happy when they get temporarilyDepowered at the start of the New Power arc (and is downright pissed when the titular power-up shows up, even getting an Oh Crap face when she sees the incoming power), staying in only out of her sense of responsability, and in the final story arc she's the one who wants to quit instead than facing the tasks to stay Guardians and acquire another power up.
Spider-Man. He gets asked to stop a thief and literally refuses the call; his uncle winds up paying the price. He quits a number of times as an ongoing refusal of the Call, only to find out that the Call has unlimited minutes.
JMS' Spider-Man revealed that many of the challenges Spidey has faced are caused because Ezekiel, the previous Chosen One and Spider-Man's mentor figure, refused the call out of fear and selfishness. He was the one who was meant to fight evil and save New York, but because he refused to use his powers for the greater good it fell to Spider-Man. By the time of the comic he's pretty remorseful and decides to aid Spider-Man as best he can.
Inverted in Avengers vs. X-Men. Hope accidentally Missed the Call when she panics at the initial bonding of the Phoenix Force, leading to Iron Man accidentally breaking the Phoenix Force and empowering five X-Men with it. Near the end of the story, Hope confronts one of the last remaining X-Men with it, Cyclops, and tells him she's ready. Instead, Cyclops turns her down, telling her that she had her chance and she blew it. It was his turn now.
After discovering that he has a brother (the biological one at that), younger brotherTony Stark offered an armor to Arno Stark, but was turned down. He later shows up with brand-spanking new, tank-like Mk. MMXX (that's 2020), saying that he wanted an armor; he just didn't want Tony to make it.
After the War of Kings event killed off Queen Lilandra, Gladiator refused to take the throne until the threat of civil war forced him to do so. Given that the Shi'ar empire was in ruins, and had just come out of a brutal war with the Kree, you can't really blame him.
Average Joe In Bullet Hell's Ryan Randa receives a series of texts from a stranger on his cell phone, telling him to go someplace and to have his boss give him time off from work. This understandably creeps him out, so he takes precautions to protect himself. Fat lot of good this does him when the caller in question, YukariYakumo, eventually tires of texting him and drops him into Gensokyo anyway.
The four in With Strings Attached really don't want to be bothered restoring the Vasyn and removing the curse on Ketafa, but they have to if they ever want to get home again.
A popular sub-genre of Heralds of Valdemar fics involves the comedic (and tragic) potential of people refusing to be Chosen when their Companion arrives.
In Lockout the leading man, Snow, initially refuses the call to save the president's daughter with the line "I'd rather castrate myself with blunt rocks."
In Star Wars A New Hope, Luke Skywalker expresses a wish to get off Tatooine but refuses the call due to his current situation ("Alderaan? I'm not going to Alderaan. I've got to go home. It's late, I'm in for it as it is!"). He's later forced into it when his uncle and aunt are killed, because "There's nothing here for me now." (he kinda has a point there)
Later in the same film, as the Rebels are preparing to assault the Death Star, Han Solo refuses to take part in the attack, feeling it would just be suicide. (He changes his mind just in the nick of time, though.)
A staple of action movies and, by extension, action video games. Sometimes it becomes personal, which spurs the hero into action.
In Escape from New York, the Anti-Hero Snake Plissken is offered a pardon for his past crimes if he'll rescue the President and a lost tape. He initially refuses, but changes his mind because he thinks he can use the opportunity to escape altogether.
The Golden Child. Chandler Jarrow absolutely refuses to accept that he's The Chosen One, but helps the good guys anyway because he wants to save the title character. By the end he grudgingly accepts his fate, mainly due to his love for the heroine Kee.
In The Last Starfighter, Alex Rogan finds his favourite videogame is a military recruitment tool that got him inducted into The Star League. However, whatever interest he has vanishes when the villain Xur has a spy tortured to death on video and Alex instantly demands to be taken home, now!
However, when Xur learns about him and sends assassins after him, Alex realizes that his only chance of survival is as a starfighter.
Neo literally receives a cell phone call to adventure at work from Morpheus.
Neo refuses the call by dropping it, figuratively and literally. He wimps out in escaping the agents hunting for him, refusing to go out on a scaffold, while dropping Morpheus' phone several stories below.
The above is actually an odd aversion. They all believe that The One is a figure so important to the survival of humanity as a whole that they must be protected above anyone else, which is why Morpheus sacrifices himself to let Neo escape. Neo being tricked into thinking he's NOT The One frees him into taking the call to save Morpheus without thinking he's putting Morpheus ahead of humanity.
After a dramatic buildup, Gary in Team America: World Police abruptly decides not to join Team America. Of course, after a montage sequence of visiting American monuments and listening to a hilarious satire of patriotic country musicnote "Freedom isn't freeeeeee, Yeah there's a hefty fuckin' feeee...", he eventually changes his mind.
Ardeth Bay: If I was to say to you, 'I am a stranger traveling from the east, seeking that which was lost,' what would you say?
Rick O'Connell: Then I would reply, 'I am a stranger traveling from the West, and it is I who you seek.'
Ardeth Bay: Then it is true. You have the sacred mark. [points to tattoo]
Rick O'Connell: That? No, that got slapped on me when I was in an orphanage in Cairo.
Ardeth Bay: The mark means you’re a protector of man, a warrior for God, a Magi.
Rick O'Connell: Sorry, you’ve got the wrong guy. [lightning strikes]
In Terminator 3: Rise of the Machines, John Connor refuses to be the leader of the resistance against Skynet and even at one point threatened to commit suicide in front of the T-800. This is mostly because he knows that his being leader of the resistance means that billions of people will die to nuclear holocaust. By the end of the film, he takes up this responsibility one way or another.
In both Braveheart and The Patriot (AKA Braveheart recycled IN AMERICA) Mel Gibson plays an influential but unambitious farmer who is asked by his countrymen to help in a rebellion against their oppressive king. He refuses to take up arms because he'd rather live a simple life than go to war. Then the king's soldiers kill one his his loved ones and leave him no choice.
In Serenity, Mal's first reaction to River's sudden bout of insanity and ludicrous martial-arts skill is to hide. Unfortunately, The Operative has already burned down his destination on Haven and killed everyone there, causing Mal to snap. It goes poorly for all sides. (One of the themes of the movie is that if the Alliance knew to leave well enough alone, everything would go more smoothly.)
In Jumper, David is watching the news while reporters detail severe flooding which has trapped multiple people on their roofs. The reporter even questions what will happen to them. Saved by a mysterious teleporter? Nope, he was busy banging some British chick.
Bilbo in The Hobbit initially wanted nothing to do with Gandalf or the Dwarves, simply wanting to be left in peace. But he eventually Jumped at the Call when he woke up the next morning and realized his normal life just wouldn't be enough if he didn't do it.
In the 2010 film version of Alice in Wonderland, a prophecy says that the Jabberwock (a beast that serves the Red Queen which she uses to intimidate her subjects into submission, and pretty much the one way she keeps a hold on power) can only be slain by the Vorpal Sword, and that Alice is the only one capable of using it. However, Alice spends most of the film denying that she is capable of it, going so far as to insist to the creatures who tell her this that they have the wrong person. (The film seems to be a Coming of Age Story, more or less, so Alice has to do a lot of Character Development before she comes to her senses and accepts the call.)
Dogma takes this and runs the whole nine yards. Bethany doesn't take her job in trying to stop Loki and Bartleby from getting back into Heaven seriously, despite being told by Metatron and meeting "prophets" Jay And Silent Bob, the 13th Apostle Rufus and the muse Serendipity. It isn't until they meet the two and Rufus reveals that she's a descendant of Jesus Christ that Bethany outright walks off from the group and begins ranting and raving as to why she's suddenly being thrust into all of this. Metatron shows up again to comfort her and even says that Jesus Himself pulled this when he found out about his heritage (it's hinted that he went into a Heroic BSOD when he found out, explaining the large time gap in The Bible.)
Inverted at the end of The Avengers; when Agent Hill asks Directory Fury if the Avengers will return, he says they will. When she asks how does he know, he responds, "We'll need them to."
Our Man Flint - superspy Flint is called on repeatedly by his erstwhile boss Kramden to save the world, but brushes him off disinterestedly. It's only when Kramden is hit in an assassination attempt meant for him that he figures he's in whether he likes it or not (and saves Kramden, naturally.)
In the Dresden Files novels, we find out, of all people Charity refused to use her magical gifts. In the Dresdenverse, magical ability is "if you don't use it in your youth, you lose it." Molly made the call not long before the same thing would happen, but incurred the wrath of the White Council for the type of magic used —- Mind Rape breaking one of the Seven Laws of Magic.
Matt from The Power of Five tries this in every book; unfortunately, the call either finds him or he's coerced into him.
Rincewind of the Discworld books always refuses the call, being a complete and utter coward; however, he invariably ends up being forced by the plot into following it. This is subverted in The Last Hero, where he actually volunteers for a space mission, having realised that he's bound to be chosen for it anyway.
"I do not wish to volunteer for this mission." Rincewind said.
"I beg your pardon?" said Lord Vetinari.
"I do not wish to volunteer, sir."
"No one was asking you to."
Rincewind wagged a weary finger. "Oh, but they will, sir. they will. Someone will say: hey, that Rincewind fella, he's the adventurous sort, he knows the Horde, Cohen seems to like him, he knows all there is to know about cruel and unusual geography, he'd be just the job for something like this." He sighed. "And then I'll run away, and probably hide in a crate somewhere that'll be loaded on to the flying machine in any case."
"Probably, sir. Or there'll be a whole string of accidents that end up causing the same thing. Trust me. sir, I know how my life works. So I thought I'd better cut through the whole tedious business and come along and tell you I don't wish to volunteer."
"I think you've left out a logical step somewhere," said the Patrician.
"No, sir. It's very simple. I'm volunteering. I just don't wish to. But, after all, when did that ever have anything to do with anything?"
"He's got a point..." said Ridcully. "He seems to come back from all sorts of things.
"You see?" Rincewind gave Lord Vetinari a jaded smile.
Also, Susan Sto Helit in Soul Music, Hogfather, and Thief of Time. She is so sensible that she refuses to believe in the existence of anything supernatural. Unfortunately, this proves to be difficult when Death is your grandfather, and you're the substitute anthropomorphic personification of DEATH whenever your grandfather is unable to perform his duties (which seems to happen often). Even after being thrust into such supernatural occurrences, she immediately resets to a "I just want to be normal" frame of mind after aforementioned occurrence is resolved.
All of the major characters in the Wheel of Time series have "refused the call" at least once.
He doesn't actually refuse to do anything. If he had, it would have substantially changed the second and third books, as Elena wouldn't exist.
Otah Machi in The Long Price Quartet amazingly succeeds at ditching his poet masters and running off. He later becomes Emperor, of course.
There was a TransformersChoose Your Own Adventure book where at the first choice, you could refuse the call. The adventure promptly ended, with the note that you occasionally wonder what might have happened.
Elli Winter was supposed to be the Traveler from Quillan in The Pendragon Adventure. Press offered her the position, knowing she had what it took. Elli refused, though, because she felt she wasn't up to it, and asked that the Traveler position be given to her daughter Nevva. That didn't work out so well.
Bobby himself also refused the call at first. Luckily for everything that has, does, and ever will exist, the abduction of his Uncle Press made it personal.
Bilbo Baggins in The Hobbit, but Gandalf decides his hidden Tookish nature craves adventure (and he needs him for his complicated plan) so ensures he goes anyway.
Interestingly enough, according to Tolkien's notes Gandalf himself didn't exactly Jump At The Call: when the Valar (the "gods") decided to send some emissaries to Middle-earth to help defeat Sauron, only a couple actually volunteered. The Maia later known as Gandalf was asked to go but declined, saying that he was too weak to fight Sauron. In the end Manwë, the high king of the Valar, had to order him to go.
Mercedes Lackey had Diane "Di" Tregarde, a witch of astounding power and a Guardian, which apparently means even more power and the responsibility to use it to protect people. Her grandmother taught her, but after said grandmother died Di tried to stay away from all that and live without the supernatural. This didn't work out, mostly because supernatural creatures and people could still feel her potential, and trying to ignore them met that they ran into her on their terms.
Marco from Animorphs was at first very reluctant to fight with the others until he discovers his mother is alive and under control of Visser One.
He was literally going to quit, but Jake asked him for just one mission more. It wound up being this one. This may or may not be a coincidence.
Karigan from Kristen Britain's First Rider's Call turns down Rider employment after her first adventure, choosing to become a merchant like her father. But the ghost of the Riders' founder keeps trying to summon her in her dreams, and Karigan ends up sleep-riding riding halfway to the capital city in her nightgown, with humiliating consequences. At that point, she decides she'd better give up and join the service.
In The Particular Sadness of Lemon Cake, Rose's father avoids hospitals because he fears a power like his own father's will manifest there.
Miras Vara in the Terok Nor books from the Star Trek Novel Verse. When Vara is exposed to an Orb artifact, the spirit lifeform Oralius, leader of the old gods, makes Herself known and insists Vara is the next Astraea (leader of the church and vessel for Oralius). Vara doesn't want to give up her career and her home to live on the run as leader of an outlawed faith. Knowing that Oralius will send her psychic dreams, Vara tries to avoid sleep. This is of course futile. Eventually she gives in and accepts her new destiny. She certainly makes a good effort at refusing the call, though.
In Sunshine, Rae really does not want to be a vampire-slaying magic-wielder. But she's really good at it, and the humans and Con need all the help they can get.
Dave in John Dies at the End frequently tries to escape from the supernatural shenanigans that he's roped into. Perhaps best illustrated in one scene where all the Soy Sauce-related madness starts, when his Evil-Detecting Dog runs off down the street barking at something, trying to get Dave to follow her, so Dave just gets in his car and drives off in the opposite direction.
In Daughter of the Lioness, Sarai and Dove (but mostly Sarai) are The Chosen One of the Copper Isles. Sarai eventually decides that nothing will ever change, so she elopes to Carthak, resulting in panic until Dove reminds him that she has the same credentials. This actually winds up working out to the best for everyone, since Sarai would have been a disaster as queen.
Bahzell of The War Gods receives the call in the first book, with instructions to head west. By the time he accepts the call, he is fairly close to the eastern edge of inhabited lands.
In the second season premiere of 24, Jack Bauer initially refuses to help CTU avert a nuclear bomb threat because he is still grief-stricken over the loss of his wife. A chance sighting of a mother and her child, as well as a few minutes spent in his home mulling, change his mind, and he finally decides to help the counter-terrorist unit.
He does it again in season 8, even refusing Chloe's pleas to help look into a lead regarding the assassination attempt on Omar Hassan, as understandably every time he has wound up answering the call it's usually left him in a lot worse position than he previously was in at the beginning of a season. It takes Kim's encouragement to get him to agree to help out.
In the Doctor Who double episode feature "Human Nature" / "Family of Blood", the Doctor is turned human with his memories erased, and is set to live a happy, normal life as John Smith- if it weren't for the possessed scarecrows and the insane maid who thinks he's an alien. At first, Smith refuses the call, but eventually is forced to become the Doctor again- and boy, is he angry.
In the 1996 film, Grace ultimately chooses not to go with the Eighth Doctor.
And with the finale of Series 6, answering the call for so long has made the Doctor become truly feared by many, to the point where whole religious orders have sprung up to kill him and many innocents along the way. It reached such a breaking point that the Doctor had to choose between dying to save the universe and faking his own death so the universe will forget him. As a result, it seems he may now be refusing the call after a lifetime of answering it.
In the Christmas Special "Runaway Bride" Donna, utterly terrified and drained by the events of the day explicitly refuses the Doctor's offer to travel with him. It takes a whole season for her to find him again after changing her mind, and when she does she says- on multiple occasions- that she never wants to leave.
Jaye on Wonderfalls has to be verbally harassed by talking tchotchkes for most of the pilot before she "Surrenders to Destiny".
In Mighty Morphin' Power Rangers, the chosen five get teleported to the Command Centre by Zordon and initially reject his offer to become Power Rangers, having no idea what he's talking about. It's only after a fight with evil putties that they get an idea of what actually might be going on and change their minds.
In the season two finale his refusal of the call leads to a number of unfortunate events and he runs away. It ended rather badly and he fully deserves the load of What the Hell, Hero? Chloe hits him with.
Not to mention having other characters, most notably Chloe and Oliver, yell at him to pick up the damn phone.
In seasons 8 and 9 he finally answered, even to the point where he's the one pointing out the retired members of the Justice Society that they're not returning their calls. May count as a Gone Horribly Right at the beginning of season 9 when he detaches from humanity and leaves behind a heartbroken Chloe after Jimmy dies and Clark refuses to save him.
Nathan Petrelli initially refuses to acknowledge his flight superpower (and his brother's powers that allowed him to mimic it), even when caught in the act by Hiro Nakamura in 1.5.
Hiro: "Destiny is calling!"
Ando: "I wish destiny would lose our number."
In Babylon 5 it took Sheridan a fierce dressing-down, a lot of denunciations and even a threat to his own life to drag Kosh down from the Vorlon's Monte Aloof and into the fight with Shadows. To a certain extent practically every race, short for Narns maybe, had to be persuaded, painstackingly negotiated, beseeched or coerced into action.
And, the second Kosh gets big-time involved...he's brutally murdered, as he knew would happen. Kosh wanted to delay his Mentor Occupational Hazard as long as possible
In Teen Wolf, Scott doesn't want to be a werewolf, so he ignores the call as long as possible. Unfortunately for him, the call keeps calling, so the choice ends up being: Man up and deal with your new-found werewolfness, or kill your loved ones.
In the first episode of Todd and the Book of Pure Evil, the main character utilizes the Artifact of Doom in question to gain musical ability. He is then possessed and is only prevented from destroying his entire high school by his best friend; at the end of the episode, the book is set aflame and it is forced to fly away. When one of his companions tries to convince Todd to go after the book, he says it's now someone else's problem and suggests they go smoke pot instead.
Melinda Gordon's mother in Ghost Whisperer has the gift (to communicate with ghosts) but refuses to use it.
On Medium, Allison, her daughter Ariel, and her mother, all tried at some point in their lives to suppress their psychic abilities. Allison's half-brother also does not like to acknowledge his power.
In The Bible, Moses initially refuses God's call to go back to Pharaoh's court and demand the Israelites' freedom. (His excuse: he's a stutterer - literally, his "lips are sealed." Also, the fact that he's wanted for murder might have something to do with it. God's response? Moses still has to go, but he can bring his brother Aaron along to make the speeches.)
Jonah is literally called by God to deliver a message to a city. Jonah doesn't want to do it, so he tries to run away on a ship. Of course, trying to run away from an omnipresent deity doesn't really work so much. God causes a storm to come up at sea that threatens to sink the ship. Jonah knows why the storm is happening and tells the crew to throw him overboard to save themselves. They reluctantly do so. and that's where the titular giant fish comes in. Making this Older Than Feudalism.
Very popular for Biblical prophets, actually. Everybody from Abraham ("What if they kill me to get my gorgeous wife?") to Jesus ("Okay, being executed really sucks") does it.
And it goes past Jesus, beginning from Peter ("Before the rooster sings, you'll thrice deny me") to Paul ("Saul, Saul, why are you persecuting me?")
This also featured into the founding of Islam; some historical traditions hold that Muhammad initially refused to receive The Qur'an because he couldn't read it - and later on, contemplated committing suicide, for fear of becoming a madman.
Angels In America: Prior Walter's immediate reaction to a visitation from an angel and being summoned to be "the Prophet"? "I. WANT. You to go away! I'm tired to death of being done to, walked out on, infected, fucked over and now tortured by some mixed-up reactionary angel—"
Glinda refuses the call to adventure when Elphaba asks at the end of Act 1 of Wicked. Unusual in that she refused because accepting would cost her everything, and in the end she pretty much lost everything anyway, it just wasn't due to her refusal of the call.
In Assassin's Creed II Ezio initially insists he only wants to protect his family from the main villains, preferably by getting the hell out of the country, but after seeing his uncle fight them he is shamed into helping and joining the cause.
The story of the canceled Dreamcast game Castlevania: Resurrection had a Belmont run away from his duties of killing Dracula, forcing time to rip his ancestor Sonia to his time to fill his place, although he's eventually forced to take up the whip, regardless.
In Betrayal at Krondor, Owyn is the only one who isn't compelled by duty or has a truly personal stake in stopping the moredhel schemes. He tries to go his separate way again after helping an injured Locklear who stumbles into his camp with a chained Gorath in tow. When they realize afterwards that he could talk to the wrong person and get them all killed if they just let him go, Locklear says he's either coming along with them or getting his throat slit, which would be undesirable for both of them. Later, however, when that part of the mission is done and it's assumed he'll be going home, he goes out of his way to stay with them.
Lost Odyssey's Kaim doesn't actively refuse the call, but he recognizes that, as an immortal, he has all the time in the world, and so he takes his time. But when it comes to the call of adventure that his lost memories provide, he initially refuses due to the tremendous pain he knows they will bring if he can unlock them again.
In Devil Survivor, at a specific point in the game, instead of choosing to keep fighting, you can choose to abandon everything and just try to escape the Hell on Earth scenario you're in, knowingly leaving thousands of innocent civilians to die and the city you're in to go insane and be completely destroyed. But making this choice will costyoudearly. It leads to the single worst ending of the entire game. The vast majority of your friends, along with hundreds of thousands of innocent people, die extraordinarily horrible deaths. You get delivered a staggering "The Reason You Suck" Speech by the top angel in Heaven for being a coward. And you cause an eternal Crapsack World where humanity is either completely stripped of its free will or trapped in a living hell where only the strong can survive and demons roam the world freely.
In Wario Land II, if you stay asleep long enough at the beginning, you get kicked out of your castle and get to play through an alternate storyline.
Inverted in The Bard's Tale for the PS2. The Bard does answer the call, but he refuses to be called The Chosen One, and threatens anyone who even thinks to call him that.
'EarthBound'': When Buzz-Buzz is telling Ness of the prophecy and what he must do, Pokey thinks he's one of the three boys and backs out from helping. He ends up being involved anyways, but on the wrong side.
In Phantasy Star IV, when presented with the genesis story of the Algo solar system, some time after Rune has chosen him as his champion, Chaz finds the Great Light to be an extremely hypocritical absent god figure and consciously rejects his "destiny". Once he's calmed down, he says that being thoughtlessly manipulated is the difference between the heroes and the villains. He eventually does come around, but does it for his own reasons.
In the PS2 version of Higurashi no Naku Koro ni, it's very possible to refuse the call by carefully avoiding setting off any triggers. It leads to an arc in which "Mion" appears to start going nuts, but Keiichi refuses to get involved, leading to the death of everyone with the actual Mion's death being delayed by ten years while she's catatonic.
In Dragon Age: Origins it is possible to refuse the call to join the Grey Wardens in each of the 6 possible openings, even though in many of them you need to join the Wardens to save your character's life. It doesn't really change anything though, because Duncan will conscript your character regardless of whether you want to go or not.
In the second game, Anders expresses his frustration at Mage Hawke for refusing the call to lead the Mage Underground, despite being the sort of leader they've waited centuries for.
In Golden Sun, it's possible to refuse to chase after Saturos and Menardi. This leads to a Non-Standard Game Over revealing the world drifted towards it's destruction and a bit of Fridge Brilliance, when you realize the destruction wasn't the lighthouses being lit, but the world withering away without Alchemy. (Or possibly elemental imbalance, given some dialogue in the second game)
In The Matrix: Path of Neo, it is possible to take the Blue Pill, resulting in a Non-Standard Game Over where Morpheus expresses his disappointment at the decision.
In Girl Genius, we get a character who's aware of this trope: Othar Tryggvassen, Gentleman Adventurer! He explains it to the heroine, Agatha (emphasis in original):
Othar: ...something you should remember as you fight evil.
Agatha: I told you. I'm not doing that. You can't make me.
Othar: Make you? (laughs, loudly) You ran straight at that danger without even thinking. That's who and what you are. You say you want a normal life. We all say that at one time or another. You deserve your chance at it. I'll be back in about three months — and we'll see how "normal" you are.
Agatha: He only sees what he wants to see. Which is why he's completely wrong about me.No, he's not. For once.
In No Rest for the Wicked, Perrault laughs at the idea of helping November, until she points out that he's bored.
In Mutant Ninja Turtles Gaiden, the turtles have such extreme psychological damage that all but Leo take up this trope (and even Leo has some aspects of this). Raph is on a self-destructive path of drinking and self-mutilation, Mikey has moved to the surface and is living the life of a reclusive writer, and Donnie stays in his lab, rarely interacting with anyone. While Raph is still up to a little rough action now and then, Mikey repeatedly breaks down at the thought of combat (even to save loved ones!), and Donnie, who's become disconnected from reality and a cold-blooded True Neutral (perhaps even edging toward Evil), comes within an inch of killing a young child to convince himself that he has nothing to do with the surface world anymore. A far cry from their days of teenage heroism!
Phase from the Whateley Universe. He doesn't want to be a superhero, he just wants to be a normal guy, become the CEO of Goodkind International, and live a normal life. Unfortunately, nobody really cares what he wants.
Pity he's got into more fights then most of Team Kimba combined...though, at least he has a company to run, which is nice. Marvel
Pretty much all five protagonists from the web fiction serial Dimension Heroes are uninterested in saving the world, one going so far as to throw his Guardian Band away. Unfortunately, none of them have any choice in the matter.
This is how Jonas Wharton of LG15: the resistance reacts when the Hymn of None attempts to persuade him to lead the Resistance in the first chapter, "A Call To Arms". In the end, the FBI turn up investigating claims that Jonas has committed acts of terrorism and murder, which forces him into action.
In Avatar: The Last Airbender , Aang runs away after being informed (prematurely) that he is the new reincarnation of the Avatar. He is then frozen in a iceberg and reawakens to a world in a century-long war that needs the Avatar now more than ever.
Takua tries to do this in BIONICLE: The Mask of Light by tilting the mask so that it cast its light on Jaller, making everyone else believe that Jaller was The Chosen One and sending him on the quest instead. However, Jaller manages to drag Takua along anyway.
Kung Fu Panda plays with this trope: despite his eagerness to watch the tournament in which the new Dragon Warrior is to be chosen, when Po is 'mistakenly' selected by Master Oogway, the panda protests that he can't be the one. Amusingly, but also distressingly, the Furious Five and Master Shifu refuse to accept he's their predestined hero either, and do all in their power to convince, persuade, or downright browbeat him into quitting. (And this is lampshaded by Tigress, who insists he doesn't belong and needs to get out so the real hero can show up.) Yet he continues to keep trying to learn, mostly because he's always dreamed of learning kung fu.
Once he finds out what he's expected to do, however (defeat Tai Lung all by himself), he literally runs away from the temple. (Perhaps it should be Fleeing from the Call?) Luckily for the moral of the story, Shifu stops him and convinces him he can be trained and finds a way to do it. Although Po's hometown isn't doomed, it is put in danger, and this as well as a pep talk from his Dad and a flash of insight into the Dragon Scroll convince him to accept after all.
In Danny Phantom, Danny often tries to Refuse the Call. This never works well, with the result being either just a mundane ghost attack to creating horrific alternate futures where the world is destroyed. One particular example is the episode 'Memory Blank' where a powerless Danny with no memories is forced to answer the call anyway.
In World Of Quest, the titular Quest spends most of the first episode actively refusing the call. Actually, more like kicking the call in the butt. Eventually he's tricked into accepting.
Ultra Magnus in Transformers: The Movie initially believes he is "not worthy" to keep the Matrix of Leadership and become Autobot leader, but Optimus Prime convinces him to do it. At the end of the movie, Hot Rod gets it.
In "The Burden Hardest to Bear", Rodimus Prime has been growing sick and tired of "being responsible for the preservation of the universe and its outlying suburbs", and takes the opportunity to leave the Autobots when he loses the Matrix. He's eventually Resigned to the Call, though, as he comes to realize that as The Chosen One, the Matrix is intertwined with his own destiny. And then destiny punches him in the face a few episodes later with "The Return of Optimus Prime".
When Optimus apparently came back from the dead before coming back for real Rodimus immediately returned the Matrix to him and was overjoyed when he reverted to Hot Rod. He had to take the Matrix back when Optimus turned out to be a case of Came Back Wrong.
Smokescreen of Transformers Prime has almost the exact same reaction as (G1) Ultra Magnus. Despite often speaking of his belief that he was destined for great things earlier on, he adamantly states that he doesn't think he's worthy of replacing Optimus. He even decides to Screw Destiny by using the last of the Forges' energy to resurrect Optimus.
Batman Beyond goes so far as to have a two-part episode titled The Call, in which Terry is invited to join the Justice League. In the end, he refuses to do so, at least as a full-time member.
The South Park episode "Towelie" runs on this. Being told to exchange Towelie to get their "Okama Game Sphere" console back, they end up tied up in a convoluted conspiracy plot; despite this they are totally uninterested in whatever unfolds:
Scientist: Thank you for bringing him to us, boys. You see, this is not an ordinary towel. He is the RG-400 Smart Towel, designed with a computer chip inside the terry cloth Stan: We don't care. Scientist: You see, here at Tynacorp, our goal was to make the perfect towel. A towel that would sense how wet or dry the user's skin was and fluff itself accordingly. Stan: Dude, we don't care. Scientist: Towelie was our greatest success. Smart enough to beat the average human at chess and absorbent enough to soak up even the toughest spills. But then one day, Towelie got high and just sort of wandered off. Stan: We... don't... care.