In Fullmetal Alchemist, when Edward Elric's brother sacrifices himself, the person who caused Al to do that has some serious problems. This is after many an Al-related violent encounter. Don't mess with Edward Elric's family. Don't.
During the Knightfall Saga, the Scarecrow sprays Batman with one of his fear skulls and forces him to relive the memory of the Joker killing Jason Todd. Batman, who has already run himself ragged capturing many other criminals that night, responds with rage by knocking out the Scarecrow with a single headbutt and beating the Joker (who's partnered up with Scarecrow) within an inch of his life, calling Jason's name all the while.
In DMZ a military commander that main character Matty Roth hates taunts Matty at one point by teasing Matty about his dead reporter girlfriend.
For both her parents and Chase, never bring up Gert's sad fate from Runaways. Chase gets violent. Her parents start a small scale genocide.
Films — Animated
The Lion King: when Scar thinks that Simba is at his mercy he tries to twist the knife by whispering: "I killed Mufasa.".
In Epic, Mandrake does not take kindly to Dagda's death.
Films — Live-Action
Subverted in the 2011 version of Fright Night. Jerry maliciously tells Peter "You have your mother's eyes. And your father's aim." Unfortunately Peter barely manages to get a couple of shots off at Jerry before things go to hell.
In James Bond movie Quantum of Solace, Camille does it after the general says "I knew your family, I must have been the last person to see them alive..."
Memorably used in Gladiator, except with less of the beatdown effect and instead just provoking Maximus' Implacable Man. Commodus cements his Moral Event Horizon by casually bringing up Maximus' wife's gang-rape and the crucifixion of his son while Maximus is surrounded by the Praetorian Guards. The facial expressions of the Praetorians don't budge, but they allow Maximus to leave without Commodus' permission and give him a small nod of respect as he goes. And then in the final Duel to the Death the Guards refuse to give Commodus a sword after he gets disarmed by Maximus.
And with Anakin's motives for turning to the dark side in the first place explored in the prequels, it also counts as a Not So Different moment; it was the fear of losing his only remaining family (wife and child for Anakin, sister for Luke) that drove both of them.
Anakin slaughters to death an enitre Tusken village after seeing what they did to his mother.
When a young Bruce Wayne confronts mob boss Carmine Falcone in Batman Begins, he casually relates Joe Chill's highly untrue version of how Chill killed Wayne's parents for little purpose other than to Kick the Dog as his mooks are about to drag Bruce out of the building.
In Robin Hood: Prince of Thieves, Guy of Gisbourne taunted Robin about his father's death so that Robin would leave the cover of Sherwood forest and try to fight Guy and all his mooks, something which would certainly get Robin killed. He might have succeeded had Azeem not stopped Robin and talked sense into him.
The Kurgan from the first Highlander movie does this twice, once intentionally, once unintentionally. The last time he runs into Connor before their final duel, he brags about killing Connor's mentor Ramirez, and, obviously pleased about himself, also gloats about raping Ramirez' woman. At the time that he said that, the Kurgan didn't know that that woman he raped was actually Connor's wife Heather, but once he saw the look on Connor's face and realized the truth, he quickly turned it into a Post Rape Taunt as another way to get under Connor's skin.
This occurs in Harry Potter and the Philosopher's / Sorcerer's Stone, when Harry, upon meeting Voldemort for the first time, is taunted that he will die like his parents, that they died "Begging [Voldemort] for mercy." Not quite the same, but it gives Harry strength.
Insulting Harry's parents is a very effective way of getting him to fly off the handle. Pushing that button was, after all, Aunt Marge's big mistake.
In A Dance With Dragons, Stannis' army gets caught by an immense snow storm in the middle of marching. Bogged down for weeks, some soldiers resort to cannibalism to survive. When caught, they are sentenced to die by burning at the stake. One of the soldiers involved intentionally taunts the executioner about eating the executioner's cousin so that he will be killed quickly by the executioner (who slits his throat) instead of burning alive.
Live Action TV
One of the relatively rare cases of a hero doing this to a villain shows up in Season 3 of Buffy the Vampire Slayer. The Mayor, the Arc Villain of that season, has basically adopted Buffy's Evil Counterpart Faith as a daughter, and shows true affection for her. Buffy has to fight with Faith and puts her into a coma during the course of the season. After the Mayor has turned himself into a giant snake demon in the season finale, Buffy taunts him about what she did to Faith in order to get him to chase her directly into a trap set by the good guys.
Arl Howe will taunt the Human Noble Warden this way in Dragon Age: Origins. Depending on whether you're playing as The Hero or a Villain Protagonist, it could be either played straight or subverted (he dies either way, but it could be you don't care about what he did to your family).
As part of Nale's backstory in The Order of the Stick, he once tried to usurp power from his father. During the conflict, Nale killed several children of his father's friend/adviser Malack. When the two run into each other again, Nale's first comment to Malack is "Malack. How's the family?" Malack goesberserk.
In Mario Brothers, Mario was already sworn to protect the princess, but Luigi's death at King Koopa's hands made Mario even more determined to see Koopa's defeat.
In Spain, a man on parole from his sentence for raping a young girl saw the girl's mother and taunted her by saying, "How's your daughter?" She went home, got a can of gasoline, and set him on fire.
Evidently, contract killer Richard Leonard Kulinski, also known as "The Iceman" was extremely protective of his family. Between this, and the fact that he was, yanno, a hitman, people who threatened said family tended to disappear. Violently.