X Must Not Win

"Rarity: Oh Fluttershy, would you be a dear and fly faster please!

Fluttershy: I can't!

Twilight: If you can't catch [Rainbow Dash], Discord wins!

Fluttershy: That big... dumb... MEANIE!"

- This scene from My Little Pony Friendship Is Magic

This is when wanting to avoid letting a particular character be victorious provides fuel for the other character's persistence in fighting against them.

Of course, it is a somewhat spiteful component of persistence; and even when it's the good guys who feel this way about the bad guys, it does arguably raise some form of moral ambiguity that this fuels it; but expect more child-friendly works to gloss over this. On the otherhand, this is also frequently used in cases were the character in question winning really would be a very bad idea.

See also Spiteful AI. This is one of the main drivers of political maneuverings both in fiction and in Real Life. See Enemy Mine, Balance of Power, etc.

Anime and Manga

 * Parodied in Sket Dance. Telling the Student Council or the Sket Dan that the other group is involved in something is a perfect way get to them to join in. This is especially pronounced with Tsubaki and Bossun, due to

Film

 * Batman's insistence in The Dark Knight on not letting The Joker win.
 * Disney's The Hunchback of Notre Dame has the gargoyles trying this on Quasimodo,

Live Action TV

 * In the Father Ted episode "Cigarettes and Alcohol and Rollerblading" after being goaded into giving up something for Lent by Father Dick Byrne, Ted gives Dougal a lecture on the importance of Lent, something far more important than the sacrifices made by Jesus as the latter points out, but beating Dick Byrne at his bet.

Tabletop Games

 * This essay on the Kingmaker Scenario calls this "The Vendetta" - whatever the reason, player A no longer cares about trying to win the game themselves, and has adopted an attitude of "I'm going to do everything in my power to make sure player B doesn't win."

Western Animation
"Butters: Whoa, you sure seem with it, Eric. You must have some... ih-inspiration.
 * The My Little Pony Friendship Is Magic two-parter "Return Of Harmony" has
 * Plankton of Spongebob SquarePants is a conniving sociopath who usually wants to steal the Krabby Patty formula and take Mr Krabs' hard earned success. However, even when Plankton is using legitimate methods to gain customers, Krabs is insistant on sabotaging any profit he makes and claiming it for himself. This reached such obsessive grounds that Krabs became obsessed with destroying Plankton when he gained one customer, making it not just a case of "X Must Not Win" but "X Must Not Lose Humbly".
 * Kyle's feud with Cartman in South Park is usually justified given the latter's highly malicious intent. However even in petty wars, Kyle takes a sometimes disturbing extra mile to make sure Cartman's plans go up in smoke. eg. In "Douche And Turd" he and Cartman create opposing school mascots, with Kyle using increasingly manipulative methods to get people to vote against Cartman's candidate. Similarly a lot of Cartman's intentions are built around making Kyle lose.

Cartman: Yes, the tears of Kyle Broflovski when he loses his ten dollars to me."


 * To add onto the Cartman example, there are plenty of occasions Cartman comes out with amazing success and fame due to a bet with Kyle, but brushes it all off because he didn't win the bet in particular. In "Christian Rock Hard" for example, Cartman bets Kyle he can make a Platinum album before him. Cartman succeeds in making a highly successful Christian rock band, gaining enormous popularity and wealth. However once he finds out that Christian record labels don't give out Platinum albums (thus he can never win his bet with Kyle) he flies into a rage in public, destroying the band's career.
 * And for more Cartman, in "Fat Butt and Pancake Head," Cartman pretends he (or his hand, at least) is Jennifer Lopez. "Ms. Lopez" creates a hit album, gets affectionate with Ben Affleck, and enjoys the wealth and fame of celebrity life via Xanatos Speed Chess. The reason is to make Kyle admit the possibility that the hand is an independent living being from Cartman and calls the whole thing off when Kyle makes even the tiniest admission.
 * In Peanuts, whenever Charlie Brown has any real chance of winning something, someone has to be around specifically to prevent him from achieving the victory, usually Snoopy.
 * The most prominent case is in A Boy Named Charlie Brown, where he is one of the two remaining contestants on a winner-takes-all national spelling bee. Charlie Brown screws up spelling "beagle" due to a combination of Snoopy (who is a beagle) following him along and worry over Linus getting angry at Charlie Brown for a trivial reason.
 * In Race for Your Life Charlie Brown, Charlie Brown's team is set to win the river race, but the bullies sabotage his boat, allowing Woodstock to win.

Other

 * Often times PETA's involvement in a cause will lead many people to oppose the cause more than they would otherwise, simply because they despise the thought of allowing PETA victories.