In general, if he chose to, Superman could solve every problem on the planet if he risked a Zero Approval Gambit or just forced everyone to do the right thing. But he chooses not to because he doesn't see himself as any greater than the earthlings he protects. He will gladly sacrifice himself or let a God of Evil like Darkseid escape justice if it means he can save just one human life.
This was combined with Super Dickery in the classic The Death of Superman story. Superman is chasing Doomsday (an unstoppable juggernaut that can level cities in seconds flat) while a family is trapped in the burning ruins of their house. The issue ends with Superman intentionally blocking out a boy's pleas for help to continue his pursuit. The next issue, however, opens with Superman stating that he was hoping one of the Justice League members would have woken up from their Doomsday-delivered ass-beating. He's about to turn around and go help the family anyway when Doomsday sucker punches him. Luckily, the League does wake up. In the follow up arc, the boy feels terribly guilty, and wonders if Superman would still be alive if he hadn't tried to get his attention.
In the animated adaptation it becomes a chilling case of OOC Is Serious Business. Superman rescues a cat from a tree as per usual but proceeds to deliver an ominous almost threatening lecture about how he can't be wasting his time with little things like this.
This trope is actually one of the things he looks up to Superman for. Superman is far more of an idealist than Batman is, so Batman usually takes the role of Poisonous Friend in relation to each other. But even on his own, Batman qualifies as this archetype. As the richest man on the planet, he could potentially solve even more problems than his friend if stopped being Batman for a significant amount of time and put all of his effort into running his company, but that would mean that somewhere, there's a call for help that's being unanswered.
One strong example of this trope is in Batman: Arkham City, where Talia is taken hostage by The Joker while Hugo Strange is executing a citywide "purge". Batman calls Alfred for back up and calls Oracle to get a fix on Talia's position, but both of themrefuse to help him. They flat out force him to skip saving the girl and save the city at large.
Captain America, more often than not. He thinks more like a soldier than your typical superhero, though he puts The Men First and is quite unwilling to sacrifice anyone.
Spider-Man is usually this. An example comes in Ultimatum, where Ultimate Spider-Man stays behind in Manhattan, rescuing any civilians drowning from the flooding of New York. Virtually every other hero quickly left New York to fight Magneto and Doctor Doom and save the world. Spider-Man was diving fathoms deep into the flooded streets of New York just to find one person to save and then diving right back down to find another.
Originally, this was what separated Reed Richards from his Arch-Nemesis, Doctor Doom. In most Alternate Timelines in which Doom has conquered the Earth, he eventually creates a Utopia. However, he initially creates it through fear, tyranny and sometimes outright atrocities. Reed refuses to take this method and usually acts heroically even when he knows it's logically unsound (sometimes at the coaxing of his wife). However, later depictions of Reed have abandoned this outlook.
In The Avengers, he's the one on the ground protecting civilians and directing the law enforcement while everyone else is fighting waves of aliens or the Big Bad himself.
In Terminator 2: Judgment Day. When his mother turns Vigilante and decides to kill the man responsible for the foretold Robot Apocalypse, John demands that the Terminator take him to stop her, even if his death could save billions.
Terminator: This is tactically dangerous.
John: Drive faster.
Terminator: The T-1000 has the same files that I do. It knows what I know. It might anticipate this move.
John: I don't care. We gotta stop her.
Terminator: Killing Dyson might actually prevent the war.
John: I don't care! Haven't you learned anything yet? Haven't you figured out why you can't kill people?
In The Dresden Files, this is a big part of how Harry Dresden's Chronic Hero Syndrome manifests itself and gets him in trouble, to the point that his epitaph reads "He died doing the right thing." It's most noticeable in the third book, when he's forced into a Sadistic Choice between stopping a vampire from sacrificing an innocent girl and starting an all-out war between two of the most powerful magical factions on earth. He chooses to save the girl, and the resulting war is a central plot element for the next eight books in the series.
Explicitly stated to be the duty of a Samurai in The Hagakure: Book of the Samurai. Amongst other virtues, a samurai is supposed to always act for the sake of justice without any hesitation (making a decision "within seven breaths"), and to never show any fear of death. Thinking of the consequences of their actions, and thus failing to act, is said to be shameful. However, the "heroism" here is relative—samurai were also expected to obey and serve their masters no matter what, even if they found the task objectionable.
Doctor Who: A repeated weakness of the Doctor. Threatening an innocent is an almost 100% foolproof way for a villain to protect themself from him. Stated most explicitly in an episode of The Sarah Jane Adventures when the Doctor opts to save Clyde and Rani rather than chase the Big Bad, whose Dragon (who purposely put Clyde and Rani in danger because she knew it would happen) remarks "when given the choice between saving the universe and saving the children, the Doctor goes the wrong way." However, the Time War, in which the Doctor intentionally wiped out all the Time Lords and Daleks in existence to stop them from destroying the universe, is a notable aversion on his part.
A number of times during Matt Smith's run, he's stated that if he pretty much didn't do this, then he wouldn't be able to call himself The Doctor. In "The Beast Below", since he can't figure out how to Take a Third Option, he's reading to lobotomize a peaceful, intelligent creature that's been enslaved since it was in pain and he believed freeing it would kill the surviving population of Great Britain.
Revolution: Charlie Matheson is prone to this. "The Dark Tower" had her convincing her mother Rachel Matheson that they can try to save Nora Clayton and turn the power on at the same time.
Dungeons & Dragons' "Book of Exalted Deeds" (a 3.5 Edition supplement) explicitly states that it's the duty of a Lawful Good character to never do an evil act. Any evil act. Period. The rules state unambiguously that ends never, ever, justify the means, no matter how small the evil was and how beneficial the results were.
Both Champions and DC Heroes stated that heroes were expected to rescue innocents even if it meant letting the villain(s) get away.
It must be noted: Despite providing the page quote, Mordin Solus of Mass Effect is Not an Example of the trope, though he describes it succinctly. Mordin is perfectly fine with making sacrifices if necessary, but he simply believes he made a mistake before and sacrificed too much.
Its fully possible to play Shepard this way, especially a Paragon Shepard. Give a soldier a proper burial instead of using the corpse for research, send allies to help hotspots and take more fire without them and making sure civilians have room despite shortages are all possibilities. Thanks to the mechanics of the third game, doing some of these can make it harder to save everybody.
This is the entire gameplay mechanic that Ultima IV: Quest of the Avatar resolves around. The Player Character is forced to commit performing small amounts of good and avoiding bad deeds in order to fulfill the Eight Virtues. Future games, especiallyPagan, deconstruct the hell out of this trope.
Connor from Assassin's Creed III will always fight the Templar present at the moment, regardless of whatever "greater good" that Templar claims to be pursuing.
In fact, he frequently gets in trouble with his mentor about his apparent shortsighted view of things. His mentor advocates secrecy, stealth, and operating in small degrees to affect large changes (which has been the way the Assassin's have operated for centuries), while Connor, when given the choice, prefers to take the direct path, and, at one point, advocates approaching Washington directly to inform him of the Assassin/Templar conflict, despite being told that it would be disastrous.