I Feel Guilty - You Take It

A character wins a contest or competition, and feels that the victory was undeserved for some reason, such as:

The "winner" broke a promise or other personal principle to win
The "winner" knows that the contest was biased or downright rigged in his favor
The "winner" knows that his opponent was hobbled by circumstances beyond his control
The "winner" made a mistake that should have cost him the competition, but the judges missed it
The "winner" benefited from some random event that made it a matter of luck rather than skill

When this happens and the character finds out, they immediately have a sudden burst of conscience and doesn't accept the prize or gives it away (usually to the competitor he considers to be the "true winner").

Examples of I Feel Guilty - You Take It include:

Anime and Manga

Double Subverted in Air Gear; Ikki wins a contest against Inuyama, leader of the Rez-Boa Dogs gang. However, he learns that claiming their emblem will lead to the gang disbanding. He tries to give it back, claiming that it hadn't been a fair contest, but Inuyama is too proud to accept it. Ikki reluctantly decides to keep the medal, until...

Inuyama: "Well... *ahem*... hmm. I mean... *ahmph*... If you absolutely insist then... *ahem* It's not as though I can't give it... some more thought... *ahem ahem*...

Subverted in the first season of Pokémon. Ash refuses to accept the Boulder Badge from Brock because the emergency sprinklers had accidentally gone off during their battle, which weakened Brock's Onix enough for Pikachu to defeat it. Brock insists that he still lost, however, and gives Ash the badge anyway.

Film

Saturday Night Fever: Tony Manero and his partner win a dance competition, but he thinks another couple was better and only lost due to being Puerto Rican, so he gives them the top prize.

Literature

Harry Potter and the Goblet of Fire:
- Harry offers the Triwizard Cup to Cedric Diggory, because his entrance in the tournament was unfair (and against his will). The two of them spend some time arguing about this trope, since they'd both helped each other out and had just survived a run-in with a Giant Spider through teamwork. They end up taking it together, leading to the book's real climax..
- Harry gives the prize money from the tournament to Fred and George Weasley so they can start up their joke shop with it.

Live Action TV

Scrubs: In the Season 2 Episode "My Big Mouth", Turk only gets to go to Mexico with Dr. Kelso, because He's a guy, and Bonnie isn't. So, feeling guilty, Turk gives the prize to Bonnie, and she turns him down, refusing his "charity", Since he isn't the rightful winner, he doesn't want to keep it so he gives it to Todd in the end (which also serves as a pseudo-punishment for the sexist Dr. Kelso who passed over Bonnie due to her gender and now has to put up with the obnoxious Todd for the weekend).
Community: In the episode "Modern Warfare", Jeff gives his prize to Shirley so she can spend more time with her children. The whole team had agreed to this already (since she had the best reason to need a flexible schedule) except him.
True Blood: Sookie gets a big inheritance—but A) she never liked that relative and B) she feels responsible for his death, because Bill killed him after she told Bill her uncle hurt her. She gives the money to Jason, and Hilarity Ensues.

Western Animation

The Proud Family: La-Cienega and Agatha enter an American Idol-style Music Reality Show, sponsored by Wizard Kelly, La-Cienega, being prettier than Agatha, wins due to a fix by Wizard Kelly, "Beauty sells, Ugly Repels, y'all" Penny finds out and tells La-Cienega that if she doesn't make things right, she'll have bad Karma. La-Cienega gives the prize to Agatha, who, in turn, shares it with her citing fairness.
Timon and Pumbaa: Timon and Pumbaa find themselves on a farm, just in time for the "Mr. Pig" competition. For the first few challenges, Pumbaa is losing miserably, so Timon decides to cheat for Pumbaa and make the reigning Mr. Pig fail the rest of the challenges. When Pumbaa finds out, he admits defeat and gives the pig crown back to its rightful owner.