In Code Geass, Lelouch has one of these moments right after Euphie ended up going on an unintentional killing spree because of his Power Incontinence, when a dying old lady asks him as Zero to save them.
In the Fishman Island arc of One Piece, none other than Luffy himself has one. Specifically, Jinbei asks him to be the "hero" of Fishman Island, to which Luffy informs him that he's a pirate, not a hero, and has no desire to be one. Luffy then gives an analogy about how a hero would share his meat with everyone, while a pirate would simply feast on it. He ends up agreeing to help anyway when Jinbei promises to give him some meat.
Note this is a case of 'they are really not heroes.' The Straw Hats, and Luffy in particular just do what they want, always, regardless of the consequences. Most of the time, because they're nice people, this ends up being 'beat up the bastard who made my friend cry, who just happens to be the man who's oppressing this entire region, so I have subsequently liberated a country from tyranny.' But it's equally likely to be 'try to break my brother out of prison, and subsequently release thousands of dangerous criminals who will go on to hurt people out into the world.' Luffy never forgets that he is a pirate, even when his actions happen to benefit people, so he discourages heaping praise on him, unless it comes in the form of an awesome meat-themed party.
Jack from Pandora Hearts is very insistent that he's not a hero. It only ends up making him appear humble and sweet, not to mention more tragic and cute.
The catchphrase of Izumi Curtis, from Fullmetal Alchemist. Even though she's the one who trained the Elric brothers in martial arts and alchemy, she manages to make a Badass Boast out of declaring that she's (merely) a housewife.
It gets subverted near the end of the Brotherhood anime when she gleefully exclaims she's an alchemist.
Ray Palmernote Ray Palmer is the Secret Identity of the Atom. His superpower is the ability to shrink.: Have you ever had a sinus headache? So torturous you thought your head would explode? What if I shrank to microscopic size, entered your skull, then began to grow. Imagine how that would feel.
Killer Moth: You're heroes, you don't act that way.
Finesse: Remember why we were recruited to Avengers Academy? Not because we were the best, or the most heroic. Quite the opposite. Because we were the most at risk of becoming criminals. Say hello to the bad guy.
Kairi Niko in the Knights of the Old Republic Fanfic Destiny's Pawn adamantly insists that she's "just a translator." Even after the Jedi all but shove a lightsaber into her hand, she's done some heroics looking for the Star Maps, and assembled the Ragtag Band of Misfits, she still considers herself Bastila's aide. Queue massiveHeroic BSOD when the truth comes out...
Daemon: "Why can't you be a good boy and give up!"
Chaos: "I'm a Virus, Daemon. I'm not a good boy."
Also played in Harry Potter fanfic "The Apothecary Knows Best". Severus Snape claimed to be just an apothecary when Harry called him a hero for having defeated a "Jester". (This fanfic's Dark Lord has his followers known as "Jesters" instead of "Death Eaters". Yes, this fanfic's Dark Lord isAlbus Dumbledore. Long story)
In the Lord of the Rings fanfic Left, Sam tells Frodo about his fight with Shelob very matter-of-factly and he is surprised by how his master keeps harping on it.
A variation of this occurs in in Spy Kids 3D. After exiting the video game, Arnold's, Rez's, Francis's, and Juni's respective secret, non-virtual identities. Various types of this trope occur in the conversation that ensues:
Arnold: So, who are you out here in the real world?
Francesca:(to Juni) Special agent, we need you.
Juni: I'm Juni Cortez.
The above declaration by Juni is also a reference to his facade as "The Guy," which was revealed shortly before the group's exit from the game.
Near the end of Pulp Fiction, when Pumpkin and Honey Bunny are robbing the coffee shop, and the manager attempts to cooperate:
Pumpkin: I don't know, Honey Bunny. He looks like the hero type to me!
Honey Bunny: Don't take any chances. Execute him!
Manager: Please don't! I'm not a hero. I'm just a coffee shop...manager...
Naturally, his character according to the credits was "coffee shop."
Of course, the robbers' definition of "hero" is "anyone stupid enough to make trouble for us." Criminals are funny like that.
Also this monologue by Jules, sort of:
Jules: There's a passage I got memorized. Ezekiel 25:17. "The path of the righteous man is beset on all sides by the inequities of the selfish and the tyranny of evil men. Blessed is he who, in the name of charity and good will, shepherds the weak through the valley of the darkness, for he is truly his brother's keeper and the finder of lost children. And I will strike down upon thee with great vengeance and furious anger those who attempt to poison and destroy My brothers. And you will know I am the Lord when I lay My vengeance upon you." Now... I been sayin' that shit for years. And if you ever heard it, that meant your ass. You'd be dead right now. I never gave much thought to what it meant. I just thought it was a cold-blooded thing to say to a motherfucker before I popped a cap in his ass. But I saw some shit this mornin' made me think twice. See, now I'm thinking: maybe it means you're the evil man. And I'm the righteous man. And Mr. 9mm here... he's the shepherd protecting my righteous ass in the valley of darkness. Or it could mean you're the righteous man and I'm the shepherd and it's the world that's evil and selfish. And I'd like that. But that shit ain't the truth. The truth is you're the weak. And I'm the tyranny of evil men. But I'm tryin', Ringo. I'm tryin' real hard to be the shepherd.
In Team America: World Police, Gary decides "I'm not a hero! I'm a dick!" after a mission results in the deaths of innocent bystanders. A drunk he meets at a bar gives him a pep talk about dicks, pussies and assholes that makes him feel a little better.
In Jumper, Griffin spits this at Davy as he walks off to what Griffin believes to be certain death; in his experience, Jumpers don't have the power to save anyone, as while the Paladins are ready, willing and able to kill entire families as collateral damage in killing Jumpers, all Jumpers can do is Jump away. The best Griffin's ever accomplished is escaping with his life after killing his enemies, and he's been Forced to Watch everyone he's ever cared for die at the hands of the Paladins.
Griffin: You're not a hero! You're a Jumper! Don't you get it? You don't save the girl! You lose her!
Jack Reacher, in his self-titled movie, makes it clear enough for even the slowest-witted villain:
Reacher: "You think I'm a hero. I'm not a hero - I'm a drifter with nothing to lose. You killed that girl to put me in a frame-up, and I mean to beat you to death and drink your blood from a boot!"
In Gone, this attitude plays a huge role in the character of Sam, the Designated Hero of the FAYZ. Phrases reminiscent of the trope are consistently repeated throughout the books.
In Twilight, Edward uses this trope as an attempt to warn Bella to stay away:
What if I'm not a superhero? What if I'm the bad guy?
Harry: Stuff like that always sounds cooler than it really was. I've been trying to tell you that for years.
Mat "I'm no bloody hero!" Cauthon from The Wheel of Time, usually said after he does something heroic. He is the variety that thinks he can ignore the call, even though he's actively doing heroic things. Doubly obtuse, since he has gifted memories of hundreds of people throughout time - if there's one person who could objectively judge him a hero, it should be him.
Perrin "Stop calling me Lord" Aybara.
The Dragon Reborn stopped being this and pendulumed into Designated Hero syndrome, impressive because he is a bona fide, prophecy certified, world changing hero. He came around to a true, self-admitted hero in the final few books.
Mogworld by Ben Croshaw, Jim's not a hero he's a protagonist.
Giles: ...[Buffy] couldn't take a human life. She's a hero, you see. She's not like us.
*Giles smothers Ben to prevent Glory from ever returning*
In Doctor Who, Donna Noble frequently puts herself down by saying "I'm nothing special; I'm just a temp from Chiswick". As it turns out, she's destined to become the Donna-Doctor to stop the Daleks from exterminating reality.
Done again in "The God Complex". In order to save Amy Pond, The Doctor must break her faith in the idea of him as a hero.
The Doctor: I stole your childhood and now Iíve led you by the hand to your death. But the worst thing is, I knew. I knew this would happen. This is what always happens. Forget your faith in me. I took you with me because I was vain, because I wanted to be adored. Look at you, glorious Pond. The girl who waited for me. Iím not a hero.I really am just a mad man in a box. And itís time we saw each other as we really are. Amy Williams. Itís time to stop waiting.
In the DS9 episode "Empok Nor", Garak (under the influence of drugs) has been twisting the knife about Miles O'Brien's participation in an incident which occurred while he was serving in the Federation-Cardassian war under Captain Maxwell; O'Brien led an attack on a Cardassian encampment and was dubbed "the Hero of Setlik III".
Garak: Maybe it's true... maybe you're not a soldier anymore.
O'Brien: You're right. I'm an engineer.
This doubles as a Moment Of Awesome, when O'Brien then saves himself by activating the improvised bomb that his soldier instincts had called for and his engineering skill had built.
As Richard Alpert attempts to convince her to join "Mittelos Bioscience", Juliet Burke, who is going through something of an emotional breakdown, tells him: "Whatever you think I am, I'm not. I'm not a leader, Mr. Alpert. I'm a mess."
In the Firefly episode "Jaynestown", criminal/mercenary Jayne is considered a hero by the locals after dropping stolen money on them a few years before. He only dropped it to lighten his damaged aircraft in order to escape capture... and he threw his partner out first. He has a change of heart about basking in their adoration:
Jayne: You think somebody's gonna just drop money on you? Money they could use? Well, there ain't people like that... there's just people like me.
Once Upon a Time: When August (Pinocchio) tries to convince Emma that the curse trapping Storybrooke is real, she refuses to believe it and says that if the town is relying on her, "[they're] all screwed."
The Dark Knight soundtrack has a piece titled "I'm Not A Hero" and correlates with the scene in which the page quote is delivered.
Daniel Balavoine's song "Je ne suis pas un hťros" (I am not a hero) repeatedly states that pop singers should not be considered as heroes by their fan base. But he was not only a famous pop singer, he also was a musician and a composer, he was quite smart and some of his lyrics were like poems. And he donated a lot for charity, he stood against racism, and died young. If he was not a hero, he was close to be one.
"I'm not a hero! I'm not a savior! Forget what you know/I'm just a man whose circumstances went beyond his control." Styx's "Mr. Roboto" definitely fits. "The modern man/Who hides behind a mask" indicates this is probably the Tomato in the Mirror usage.
Josh Groban's "In Her Eyes:" I am not an hero/I am not an angel/I am just a man...
Bruce Springsteen's "Thunder Road": "Well I'm no hero, it's understood/All the redemption I can offer, girl, is beneath this dirty hood/With a chance to make it good somehow..."
Nickelback song Hero: "They say that a hero could save us; I'm not gonna stand here and wait... And now that the world isn't ending, it's love that I'm sending to you. It isn't the love of a hero, and that's why I fear it won't do..."
The message of the Black Veil Brides has always centered around being who you are and standing up for yourself. As a result they have gathered a large fanbase that worship them. While the boys love their fans and support them through their music, they've always said that they were just guys in a rock and roll band. 'Nobody's Hero,' a song from their album 'Wretched and Divine' expresses this viewpoint. They will support you, they will help you, but in the end they are only human, not heroes.
"I can lead you to salvation yet I am just a man. I'm a loaded gun, an only son. I'm nobody's hero. I've come undone, I'm on the run. Yeah I'm nobody's hero."
For a series in which over 90% of the Loads and Loads of Characters are veteran soldiers, Metal Gear is amazingly pacifistic and permanently deconstructing the ideal of the 80s and 90s action hero. It's particularly emphasized in the first and fourth games of the main series when Solid Snake rebukes Ascended Fangirl Meryll, who claims to have trained to become a soldier all her life because she was inspired by him.
Sora: You're really good? Are you some kind of hero?
Auron: No. I'm no hero. I'm just anÖ
After which he then gives his name.
Similarly, in Final Fantasy X, when Wakka refers to Auron as a "legendary guardian," he replies "Legendary guardian? I was just a boy. A boy about your age, actually. I wanted to change the world, too. But I changed nothing. That is my story."
Megas XLR: Look, Red, I'm no hero. I'm just a guy from Jersey.
Elisa in Gargoyles (specifically in "High Noon") says she's no hero; she's just doing her job. Not that this stops her from fighting tooth and nail to save her friends when she isn't on the job...
In Jackie Chan Adventures, Jackie would like nothing more than to do his job as an archaeologist and leave the heroism to the authorities.
In The Hair Bear Bunch episode "I'll Zoo You Later," the superintendent praises Peevly as a hero for corraling two bank robbers and the runaway Bunch. Peevly says it's simply "all in a day's work."
In an episode of Batman Beyond, a Fantastic Four-ish group called the Terrific Trio goes insane and tries to set off a nuclear reactor. Terry tries to stop their leader by reminding him that he's supposed to be a hero.
Magma: No... I'm an accident. Real heroes, they make a choice. I never did.