Emiya. Shirou. Good god, Shirou. Helping people is one thing. Helping little girls out is even better. Helping little girls who were just stabbed in the chest by what is damn near a physical god, at least compared to YOU, is not an advisable course of action. And you HAVE to do it or you get a bad end.
Kenzo Tenma from Monster will never let an injury go ignored. Ever. At one point, an unscrupulous character wanted to recruit him as an underground physician, but really, that's a good part of what he was doing already. It actually turned out very poorly... because he convinced the patient to turn himself in and didn't even charge him.
In Muhyo And Roji, Roji fails a test that would allow him to officially advance to First Clerk and stay as Muhyo's assistant, when he takes too long to help carry out an injured fellow student.
Dwight in Sin City does a lot of good for the girls of Old Town despite being wanted for murder.
A minor example would be Marv protecting people while on parole. This often means lots and lots of violent things happening.
In Dragon Age The Crown Of Thorns, it is played perfectly straight by the dwarven noble protagonist, of all people. You'd think that a politician that spawns one plan after another and who has the whole world on his shoulders would be a bit less eager to go into potentially deadly situations with just his dog as help. Turns out that he doesn't think too hard on whether or not to totally go out of his way in the opposite direction (about a day's journey too) and check on Honnleath and if anyone was still alive there, even though [[spoiler:he'd finally seen the first human after roaming the Korcari Wilds for weeks, during which time he almost got killed several times, nearly dyed of starvation and refused a deal with Flemeth that she would not outline until agreed upon. And his armor was so rusted and worn that it didn't even survive the trip to the golem village, although, true enough, the demon shattered most of it. Raonar does reason that it's important to assess whether the horde really reached so far west in such a short time or if it's just a group of stragglers. One would think this is just him looking for justification, but that wouldn't fit with his Brutal Honestypolicy that he uses on everyone, including himself, meaning that he manages to play this trope AND avert Honor Before Reason altogether (after all, he had gained some nifty magical abilities to tip the scales).
Film — Live Action
Lampooned at the end of Cannonball Run, as Captain Chaos turns aside and forfeits his team's victory when a woman screams that her baby is drowning. Said "baby" turns out to be a dog, but his teammate still forgives him for losing the race.
In the 1993 film The Fugitive, while on the run, Dr. Kimble is in a hospital as part of his search for the one-armed man. He notices that a child has been misdiagnosed and will die if not given immediate surgery. He changes the orders on the kid's chart and saves his life. The problem with this? He was masquerading as a janitor at the time.
In RoboCop 3, Robo veered off pursuing some insurgents to come to aid of a squad of cops locked in a firefight with a gang of thugs.
In The Gumball Rally, the car disguised as a police cruiser loses the race when it stops to help a pregnant lady giving birth while stuck on an L.A. freeway.
In Pee-wee's Big Adventure, Pee-wee is on the run after stealing his bicycle back from Warner Brothers Studio. He makes a grand escape, lands a big jump, and is in the clear. Until he sees a fire in a pet store. He even saves a handful of snakes before collapsing outside and being picked up by the police.
Essentially how we're introduced to Clark. He's forced to Walk the Earth because he's afraid what people will do when they discover his abilities, but he can't blend in for long because of his Samaritan Syndrome. Lois flat out tells him that the only way he'll stay completely hidden is to stop saving people.
He's slightly hampered in his fight with the Kryptonians by the fact that he's trying to protect human bystanders at the same time.
Jean Valjean from Les Misérables. More often than not, his Moments involved throwing himself directly in Inspector Javert's path, rather than merely risking the same.
Discworld example: in Thief of Time, Lobsang Ludd and Lu Tze are trying to outrun a lightning bolt, knowing that if they fail, the entire world and all of history will be destroyed. Lu Tze falters. Lobsang turns back to help!. As a result, the entire world gets frozen in time. Susan sto Helit lampshades this hard when she finds out.
Harry Potter, the boy with the "saving people thing", is of course prone to this.
In Harry Potter and the Goblet of Fire, the second task of the Triwizard tournament is rescuing a hostage from some mermaids. Harry gets there first, but ends up coming up last because he waits to make sure all the hostages are saved. It's then pointed out to him that it was all part of the contest and nobody was actually in danger, but he gets extra points from the judges for heroism.
The O. Henry story "A Retrieved Reformation" is all about this trope; a safecracker escapes from jail, adopts an alias, heads west, and gets a job as a shoe salesman, and gets engaged to a bank manager's daughter. Just as a police detective shows up in town looking for someone like him, a young child gets locked in the safe. He quickly starts work cracking the safe open, knowing it'll give him away to the cops. At the end of the stories, the detective knows damn well that he's the safecracker he's looking for, but pretends to not recognize him because he's obviously reformed.
Live Action Television
The A-Team: The team is always on the run from the military, but never seem to find a reason to flee the country. Instead they travel the country helping people everywhere, sometimes escaping in the nick of time.
Burn Notice: This is a key element of Michael Weston's personality. No matter how much trouble he is in personally he will still take the time to help someone who needs his brand of help. His friends and family frequently lampshade this but at the same time they are the ones usually bringing these cases to him. At one point Michael has just survived a bomb going off right in front of him and he is still in shock but he realizes that a man is about to step in front of a bus so he saves him. He then finds out that the man is trying to kill himself because some conmen stole the money he needed for his son's medical treatment. Despite the fact that a Mad Bomber is after him Michael does not hesitate to help the man get his money back.
Farscape: While the team does try to lay low most (well, some) of the time, they do have a tendency to take detours whenever a distress beacon is sent out or they see someone being mistreated. More often than not, said interference tends to bite them in the ass...hard.
When a group of heroes is attempting to infiltrate a hospital as doctors, having a doctor along correcting the mistakes of hospital staff has to be one of the better ways to establish genuineness.
First Wave: The protagonist Cade Foster is on the run from aliens that will one day destroy the world. It doesn't stop him from helping every person with problems that crosses his path.
The Fugitive: Happened semi-regularly to Dr Richard Kimble. Despite being on the run for murder, he would go to the aid of anyone in medical need. (Of course, The Fugitive was inspired by Les Misérables.)
The Incredible Hulk: Happened from time to time in this TV show (which was loosely based on The Fugitive which was in turn based partially on Les Miserables).
LazyTown: Had a similar story where Robbie Rotten disguised himself as Sportacus and the two of them entered a race to show who's the real one. Sportacus was the fastest of the two but stopped to save someone, allowing Robbie to win. Stephanie called Robbie out for not doing the rescuing.
Merlin: Merlin lives in a world where magic is a crime punishable by death, but he is continually using it to save Arthur's life in pretty much episode, as well as save Camelot almost single-handedly more than once. By some miracle he's managed to keep it hidden thus far.
Who Wants to Be a Superhero?: Stan Lee acknowledged this trope and punished people who didn't in a Secret Test of Character. In a test to change into your super hero outfit and race to the finish line the fastest, there was a little girl crying for her mother mere yards from the goal line. Anybody who stopped to track down the girl's mother or take the girl to the nearby park station automatically got a pass, but anybody who ignored her was a front-runner for the chopping block.
Myth and Legend
In an Irish fairy tale, a girl finds a stolen bag of gold that rightfully belongs to her family and runs off with it, and the witch who stole it naturally gives chase. The girl stops to help several animals and a mill, and they in turn gratefully slow down the witch for her.
The story of the Chinese Zodiac involves the 12 animals racing for the positions on the calendar. In many versions of the story, the dragon stops to provide assistance to various people and animals in need, thus coming in fifth place when he could have easily made first.
Yu-Gi-Oh!: This trope was mentioned in a Gagaga Academia Tospedia issue. Over the course of his journey, "Bujin - Yamato" witnessed people on the surface suffering from the tyranny of the gods and acted on it, strengthening the suspicions of the Kami dwelling in Takama-ga-hara, including the Supreme Kami.
John Landy at the 1956 Australian National Championships in athletics. He was leading in the 1500 metres final. Fellow runner Ron Clarke tripped and fell, and Landy doubled back to help him up. He went on to win the race.
Fahrenheit, AKA Indigo Prophecy: You're on the run from the police, accused of a murder that you... well, you totally committed it, but you were being controlled by a thousand-year-old Mayan Voodoo priest, see? Though for some strange reason, the cops don't wanna hear about it. On the bright side, having been briefly possessed has unlocked hidden superpowers and limited precognition in you. While secretly meeting your brother in a park in the middle of the winter, you get a warning flash of danger - not to yourself, but to a kid playing on a frozen lake nearby. The ice is gonna break, and he'll either drown or die from hypothermia! With your superhuman speed and powers, you can save him... but a pair of beat cops are patrolling the park, one of them having seen you leave the crime scene - if you make a hero of yourself, he's bound to notice! Dilemma! You can actually choose not to act but it was apparently the worst thing you could possibly do for your Sanity Meter. Fortunately, after you pull the kid out of the ice, the cop rushing to the scene, despite recognizing you, decides to pretend not to notice, figuring that arresting a guy who's just saved a kid's life would be really impolite.
Not just impolite. If you save the kid, this is referenced later when the cop admits to Carla that he could have made an arrest, but couldn't bring himself to do it right after witnessing such a selfless act. Carla says she probably would have felt the same way in the other cop's shoes.
It's easier to list RPG heroes who don't follow this trope than not. It doesn't matter that Dr. Von Destruticius is going to drop a meteor on you now, some random preschooler has asked you to get their cat out a tree and you're gonna do it.
Anytime the cast from the Dungeons & Dragons cartoon show actually found a portal home, it was guaranteed that they'd have to turn around to save the character of the week while the portal closed.
In the Saturday Morning Cartoon version of Mother Goose and Grimm, the title character of the show-within-a-show Karl the Wonder Poodle is temporarily diverted from saving the person who made the original call for help (a kid who fell in the river and is floating toward a waterfall) by the need to deal with a dozen or so other events in need of a superhero.
All the time in The Zeta Project. Zeta does this so constantly that it rubs off on Ro, because no one in her life had ever displayed this kind of behavior before.
The heroes of G.I. Joe: Renegades get this with varying intensity. On one extreme is Scarlett, who is only concerned with clearing the team's good name and resists helping; on the other end is Duke who has to right every wrong he sees. The others fall somewhere between these two ends.
Cars: In the final tiebreaker race, Lightning McQueen gives up winning in order to help Strip Weathers, who had just suffered a serious crash.
In the Wander Over Yonder episode "The Fugitives", Wander and Sylvia have to escape a planet before the Watchdogs catch them, but their escape attempts keep being derailed by Wander's need to help anyone in need. Eventually, everyone whom Wander helped returned the favor by aiding their escape.