Dub Text in Digimon Tamers final episode. When the program that was used to defeat the D-Reaper forces the digimon to return to the digital world, Henry's dad, who knew it would happen, asks him if he can forgive him, and Henry sadly shakes his head. (In the original, the head shake meant "don't worry about it").
There's also Digimon Adventure 02 where Sam Ichouji does this to his little brother, Ken, after having caught him getting his Digivice without his permission. Eventually, this causes Ken to snap that he wishes for him to disappear, and his wish came true when he got run over by a car and that Sam died rejecting his apology.
Variation in Legacy. Darth Wyyrlok says that he isn't going to apologize for trying to murder Darth Krayt. Krayt replied that he wouldn't accept any apologize either. A duel to death ensues.
This is played with in the Empath: The Luckiest Smurf novel, as after Papa Smurf saves himself, Empath, and Brainy from falling into the lava pool within the volcano, Empath apologizes to Papa Smurf for having nearly endangered Baby Smurf just to get Papa Smurf to feel sorry for leaving his only begotten son in Psychelia. At first Papa Smurf is too angry to forgive Empath, but when he sees Empath bowing down and humbling himself before the village leader as if he was the Psyche Master, Papa Smurf also bows down with Empath and forgives him, promising that he will never leave Empath like that ever again.
Happens twice in the Axis Powers Hetalia fanfic What a Way to Fall (itself a a fan written sequel to the infamous Financial Crisis Gangbang). First is when North Italy tries to apologize to America and Canada for his involvement in the incidentnote Canada shoots it down because he feels N. Italy has no right to apologize when the latter was being an Extreme Doormat. Second is when England tries to kidnap the new America personification with the intention of turning the little girl into a Replacement Goldfishnote given England not only participated in the gang rape and refused to believe he did anything wrong until America's suicide, and then had a the nerve to accuse Canada of being an unfit caretaker without any evidence, it's hard to fault Canada for rejecting this apology. The chapter is even titled Sorry Is Not Good Enough.
Perhaps one of the most heartbreaking examples was in Kung Fu Panda, where during the climatic battle between Shifu and Tai Lung, Shifu apologizes to Tai Lung for allowing him to go down his path of darkness, only for Tai Lung to reject it and demand the dragon scroll.
In The Incredibles, Mr. Incredible apologizes to Syndrome for the slight that put him on the path to supervillainy. Syndrome answers "See? Now you respect me. Because I'm a threat," and continues monologuing. Mr. Incredible realizes that Syndrome can't be reasoned with, and starts throwing stuff at him.
After "totalling a mammoth" in Brother Bear, Rutt refuses to accept Tuke's apology and declares their brotherhood over, even going as far as to say that Koda is his new brother. But after a heartfelt talk between the two moose, they feel happy again and walk off.
In Pitch Perfect, Beca, the protagonist goes all the way up to her love interest to apologize for being a Jerkass earlier. The love interest curiously, doesn't accept or even appears to want one, instead pointing out her flaws in driving people away.
In the 2007 film Bridge to Terabithia, Jess Aarons tried to apologize with his younger sister, Mary Belle for pushing her because he was angry about the death of Leslie Burke and told her he misses her talking to him, only for Mary Belle to ignore his apology by turning away from him.
Trentino: I'm sorry we lost our tempers. I'm willing to forget if you are.
Firefly: Forget? You ask me to forget... Why, my ancestors would rise from their graves... and I'd only have to bury them again... A Firefly never forgets.
Trentino: I am willing to apologize... I'm willing to do anything to prevent this war.
Firefly: Nothing doing!! I've taken a lease on the battlefield. I'd lose my deposit, besides, I've already ordered the ammunition.
Vera: Oh, Your Excellency, isn't there something I can do?
Firefly: Yes, but I'll talk to you about that when we're alone.
Mrs. Teasdale: Oh, won't you reconsider.
Firefly: Well, maybe I am a little headstrong... But, you know, it's awfully hard to forget what he called me.
Trentino: What I called you... Why, what did I call you?
Firefly: I don't remember.
Trentino: (With a little chuckle) Oh — you mean... worm?
Firefly: No, that wasn't it...
Trenitno: Was it — swine?
Firefly: No... it was a seven letter word.
Trentino: (Thinking, then with a broad smile) Oh yes! — UPSTART!
Firefly: That's it...
Takes gloves from Trentino's breast pocket and socks him across the face, and puts them in his own breast pocket. Trentino becomes apoplectic
Trentino: Why - er - Mrs. Teasdale - this is an outrage! This man is impossible... My course is clear... this means war!
In one episode of The Big Bang Theory, Sheldon accidentally ruins Howard's chances to work on a space laser by revealing to a security clearance woman that he messed with the Mars rover. Feeling guilty, Sheldon apologizes, only to find out that Howard isn't so willing to forgive.
After Frasier convinces Roz to ask someone out which ends in disaster, he asks before an ambitious broadcast for her forgiveness. Roz sarcastically complies, especially, as she adds, under the thought he will out there on air humiliating himself in front of thousands.
Frasier: So when are you really going to forgive me?
Roz: When I get married!
Also happens when Frasier is caught looking in Daphne's room, realizing she is strongly protective of her privacy, which ends with him having to bribe her with a car so she doesn't quit. After he inadvertently meddles in her belongings in a later episode, it seems the process will repeat, until a furious Daphne accidentally reveals she was snooping through Frasier's own belongings in a rant. After being called out on this, a flustered Daphne decides she'll let it go this time.
In the Babylon 5 episode "Comes the Inquisitor", Vir ends up on an elevator with G'Kar, and tries to apologize for the Orbital Bombardment on the Narn homeworld by the Centauri, which occurred in the previous episode.
Vir: "I'm sorry. I wish... there was something that I could do. I tried telling them, but they wouldn't listen. They never listen... I'm sorry."
G'Kar(pulls out a knife and cuts his own hand, and as drops of blood drip to the floor, says): "Dead... dead, dead, dead, dead, dead, dead, dead, dead, dead... How do you apologize to them?"
Vir: "I can't."
G'Kar: "Then I cannot forgive."
One wonders what G'Kar would have to say about Vir's subsequent actions, as revealed in "Sic Transit Vir"...where Vir saves the lives of thousands of Narns by faking their deaths.
In the episode "Ship of Tears", when Delenn finally brings G'Kar into the Conspiracy of Light and discloses that she knew all along about the Shadows and their aid to races like the Centauri—but explained to him that consequences and casualties would have been much worse than what befell his homeworld had they took action and shown their hand against them too early—G'Kar did understand but was still angry that millions on his homeworld were sacrificed. Delenn asks if he'll ever find it in his heart to forgive her, and tearfully he replies:
In Fringe, Walter's halting attempt at apologizing to Astrid for injecting her and knocking her out is met by her completely ignoring him.
In a later episode, Peter apologizes for selling Walter's book collection while Walter was at St Clare's, and Walter gives him a brusque "Apology not accepted!"
In the Friends episode "The One With The Baby Shower", Monica forgets to invite Rachel's mother to Rachel's baby shower, and spends the whole episode desperately trying to have her apology accepted.
In Star Trek: Deep Space Nine, Odo tries making a verbal apology to Kira after he breaks with the Female Changeling, under whose influence he abandoned their resistance, leading to Rom's arrest, impending execution, and looming disaster for the Alpha Quadrant. Not surprisingly, she tells him that are "way, way past sorry." (He does earn her forgiveness for helping them fight off the Dominion and a long conversation some episodes later.)
This happens quite a lot on Supernatural for some reason, particularly to Castiel, who spends half his time doing catastrophically awful things (e.g. his Season 7 arc sees him swallowing all the souls in purgatory, becoming drunk on power, declaring himself God, and committing genocide), and the other half desperately trying to atone for said awful things. Needless to say, he can't always fix things by just saying sorry, and although the Winchesters always forgive him eventually he still gets a fair amount of apologies thrown back in his face. For instance in this scene from the Season 8 finale:
Castiel: Dean...I'm sorry.
Dean: For what?
Castiel: For everything.
Dean: Everything? Like uh...like ignoring us?
Dean: Like bolting off with the Angel Tablet and then losing it? Because you didn't trust me. You didn't trust me.
Dean: Nah, that's not gonna cut it. Not this time. So you can take your little apology and cram it up your ass.
This happens to Sam various times when he tries to make amends for starting the apocalypse, and other people (understandably) aren't particularly forgiving. It also finally happens to Dean in Season 9 when he apologizes for letting a friend die, only to be reprimanded by the aforementioned friend (in ghost form) for being self-pitying and selfish, and is told that he's going to have to actually do something to make it right, instead of just sitting around crying about it.
One Republic's song "Apologize" is based ENTIRELY on this trope.
Lisa Lisa and Cult Jam's "All Cried Out".
Both: Don't you know the hurt will cause an inferno?
Romance up in flames, why should I take the blame?
You were the one who left me neglected.
Male: I'm so sorry.
Female: Apology not accepted, led me to the broken heart you collected.
Samuel rebukes King Saul for disobeying God's commands in destroying every single Amalekite (he only spared the king and their sheep and cattle). Saul sincerely repents, but Samuel tells him that God won't accept it and He has rejected him as king of Israel. (Depending on the interpreter, the reason for the lack of forgiveness varies.)
The Bible also repeatedly warns that one day, there will be a final judgement, and by then, it'll be too late to repent.
According to every form of Christianity, there is no ministry in hell. It's too late to repent your sins and accept the love of Jesus into your heart once you're dead.
In the BioShock Infinite DLC game Burial At Sea: Part 1, an alternate version of Booker DeWitt, who turns out to be an alternate version of Zachary Comstock who ends up killing Anna as he struggles to take him from her actual father, another Booker DeWitt, tells Elizabeth that he is sorry for what happened. Elizabeth tells him that he's not sorry, but he soon will be... just before he is impaled to death by a Big Daddy.
In Devil May Cry, Trish is working for Mundus, so when she reveals it to Dante and started attacking him with Nightmare, it failed. She doesn't fight back after this and tried to apologize, only for a furious Dante to point his gun at her and give her "The Reason You Suck" Speech before leaving her behind, much to her sadness.
Heavy Rain would have this if Ethan chooses to reject Madison for not telling him that she's a journalist all along.
Ethan: You lied to me. I trusted you and you used me. Now I'll never be able to trust you again.
Saints Row 2: At Aisha's funeral, Johnny lays a No-Holds-Barred Beatdown onto the one responsible, and when they try to apologise, he punches them through a tombstone. Nothing says "Apology not accepted" like a concussion.
In an episode of South Park, Cartman infects Kyle with AIDS for laughing at him, leaving the latter obviously enraged. The principal finds out about this and demands Cartman give an apology (which he does half heartedly), while Kyle should also apologize for tattling on Cartman. Cartman seems sincerely appalled that Kyle is still livid.
One episode of American Dad! has Stan taking his family to his half brother Rusty's place, only to discover he's fabulously wealthy. Becoming jealous, Stan and his family attempt to take Rusty's place, only for them to be forcibly sent off with a threat that he'll kill Stan if he ever sets foot on his land again. The helicopter they're in then crashes and they're stranded in the desert. After going through hell and realizing that family is more important than material possessions, Stan attempts to make amends with Rusty, who found them in the desert, only for his half brother to attempt to carry out his threat of killing Stan for still being on his land (he owned a lot apparently).
In the Family Guy episode "Movin' Out (Brian's Song)", Brian wants to apologize to Jillian when Stewie revealed that he was paying half of the rent of the apartment, which prompts Brian to admit that he never wanted to move in with Jillian at all. She still rejects his apology and remains broken-hearted.
An episode of VeggieTales has Larry's actions accidentally leave the cast stranded on an island. Larry tries to apologize, but the others aren't willing to accept it. Then we all learn a lesson about forgiveness.
An episode of The Boondocks has the family going back to their old home of Chicago for a funeral. Huey sees his old best friend Cairo, who took it badly when Huey moved. At the funeral, Cairo snaps at him and calls him a "fake nigger", causing a fight to break out. Huey later goes to his house and apologizes for the fight. He offers his hand to Cairo, who takes it...and yanks him forward, headbutting him so hard that he flys off the porch. Well, that's one way to reject an apology.
In Avatar: The Last Airbender , Katara meets the man who murdered her mother, and sees what a sad, pathetic wretch he's become since then, to the point that he's even begging her to kill his own mother in exchange. She would have none of it and gives him "The Reason You Suck" Speech, letting him live with the fact that she will never forgive him for what he did to her and her family and that he's already living out a Fate Worse than Death.