Save the Villain

The hero and bad guy have had a big fight on the top of a building, which has resulted in the bad guy being knocked over the edge. He hangs there by his fingers, helplessly.

The hero is then motivated (or more cynically, contractually obliged) to attempt to save the villain's life, even putting himself in mortal danger in the attempt.

This is presumably done so that the hero can be shown once again to be noble and just. Sometimes they may strongly wrestle with the notion; the temptation is not just to let him die, but consider himself blameless for not directly causing the death.

If no one will realize that the hero could have saved him, many take it even further, thinking that letting someone die via inaction on their part is almost as bad.

A notable part of this trope is that, the villain refuses the hero's help because he can't bear owing his life to the hero or would rather cause his own death, presumably out of ego or "honor".

Sometimes this happens when the villain is facing a cruel fate at the hands of a greater evil who disgusts the hero even more, leading him to shout "Even he doesn't deserve that!" or something of the type.

If the villain does accept the hero's help even if they begged for it, more often than not, they will be complete ingrates and keep trying to kill the hero, sometimes even immediately.

In fact, sometimes the villain will use the opportunity to try to kill the hero — leaving both the hero and villain in mortal danger. Other times, the villain may be utterly bewildered as to why anyone would try to save an enemy.

On other occasions, the villain will continue pursuing their overall evil goal, but will now refuse to harm the hero out of grudging recognition of their debt.

They may even return the life-saving favor at a later point, although this may wipe the slate clean in their eyes (saying that they're "even" at that point) and make it okay for them to resume their attempts on the hero's life.

In between, the hero may say Think Nothing of It in the knowledge that the villain will indeed think nothing of it. On the villain part, a more upright antagonist (usually an Anti-Villain) may leave the hero be for the moment but warn him explicitly that "it doesn't change anything!"

In the worst of cases, the villain turns out to be an Ungrateful Bastard which leaves the hero (and the viewers) wondering why he bothered.

Some villains even use things like this as "proof" that Good Is Dumb. In more serious situations, the hero may be unable to save the villain's life but will still not let them suffer Dying Alone.

Example

 * Amos Slade and Copper pursue Tod, they get cornered and almost killed by a gigantic bear. Tod, seeing his childhood friend Copper in danger, turns back to fight the bear, also saving Slade from it during the process.
 * Tarzan had only meant to incapacitate Clayton, tying him up in vines. Clayton, however, is in an Unstoppable Rage and starts slashing through them...except for the one around his neck. Tarzan sees the problem before Clayton does and tries to warn him: "Clayton! Clayton, don't!" He doesn't listen, plummets downwards, and Tarzan zips after him. Unfortunately, he's not fast enough, and by the time he reaches Clayton, he's already hanged himself. One look at Tarzan's face and you know he really did hope to save him...
 * Aladdin seems to try to save Sa'luk from falling off a cliff at the end of their duel to the death, but fails. Sa'luk proves to be Not Quite Dead and later returns to menace the protagonists further.
 * Kiara does this to Zira when she's trying to pull herself up. Kiara offers a hand (paw?) to Zira to pull her back up, but she swipes at her and falls to her death in the raging flood below.
 * Woody and Buzz save Lotso in Toy Story 3 from a shredder without hesitation, even though it was Lotso's fault in the first place that any of them was in a life-threatening situation. Lotso doesn't return the favor
 * In the Spider-Man movies, both the original trilogy and reboot, Peter Parker frequently tries to talk the villain down, since most of them are actually friends of his, so he desperately doesn't want to have to fight them.
 * In Suicide Squad (2016), after the Joker's car falls into a river, Batman pulls Harley Quinn out and revives her with CPR before arresting her. Judging by the look on his face, he really didn't want to.
 * Guardians of the Galaxy Vol. 2, Nebula hunts down and attacks Gamora on Ego's planet once she gains a ship for herself. Gamora is able to hold her own against her sister, however, and blasts Nebula's ship. Before the ship goes up in flames, she drags Nebula out. The two reconcile later.
 * In Spider-Man: Homecoming, Peter saves Vulture when the villain becomes trapped under flaming rubble. Vulture is grateful for this and returns the favor by refusing to reveal his identity to Mac Gargan a.k.a. Scorpion.