While it can sometimes be cathartic to watch a particularly nasty villain die, the same can't be said when the villain is sympathetic. The saddest anime villain deaths tend to involve evildoers whose motivations make sense, as most people can relate to the bad thoughts that enter one's mind when pushed into a corner. If a villain was clearly birthed from extreme pain, it's hard not to feel a little sorry for them, even if they also eat babies sometimes.
Naruto practically invented the concept of tragic villain deaths. Just about every antagonist is sympathetic in some way, and almost none of them survive. However, the first character to fall victim to this trope is Momochi Zabuza.
When he's first introduced, Zabuza is working as a bodyguard for a corrupt businessman named Gato. What Zabuza really wants is to kill the Mizukage (the leader of his nation), but he needs money to continue his quest after the first coup d'état failed. Zabuza is aggressive, arrogant, and willing to kill anyone who stands in his way, but there's another side to him that's a little less in your face. Although he claims to see the orphan Haku as a tool, it's clear that he cares deeply for the boy. When Haku dies trying to save Zabuza's life, the villain is devastated, though it takes time (and a lecture from Naruto) for him to admit his true feelings.
When Gato shows up and attempts to defile Haku's body, Zabuza kills him, but dies in process. He never got the chance to act on his paternal love for Haku, or to thank him for saving his life.
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Meruem — Hunter X Hunter
Meruem of Hunter x Hunter is a half-human, half-insect hybrid who sees humans as inferior scum who are livestock at best. He's so callously violent towards them that he has no gripes eating a human baby. All that changes when he meets Komugi, a blind girl who plays a game called Gungi with incredible skill. The two forge an unexpected friendship that helps Meruem learn to respect and care about humans.
Before Meruem can change his behavior patterns, he's poisoned during a battle with Nereto. Knowing that he has no chance of survival, he drags himself to Komugi for one last game of Gungi. However, he doesn't want to infect Komugi with the contagious poison, and soon attempts to leave. Before he can make his exit, Komugi stops him, as she refuses to let him die alone. After several more games of Gungi, Meruem dies, and Komugi follows soon afterward.
Kagura is a wind sorceress who was formed from the flesh of Inuyasha's top villain, Naraku. Though she's forced to obey her originator, she hates him, and takes every opportunity she can to undermine his authority. When she finally openly defies him, he retaliates by dealing her a fatal wound.
As she's dying, Sesshomaru, Inuyasha, and the rest of the crew arrive, but they're unable to do anything to save her. Her body evaporates and becomes one with the wind, but she's happy to die because she's finally free of Naraku.
While most of the homunculi in Fullmetal Alchemist: Brotherhood meet with emotionally brutal deaths, Envy's is probably the worst. After being defeated by the uneasy alliance of Edward Elric, Roy Mustang, Riza Hawkeye, and Scar, Envy tries desperately to convince them to turn against each other. Edward realizes that Envy needs them to hate each other because he is jealous of humanity's ability to love and forgive.
When Ed correctly guesses Envy's true motivations, the villain is humiliated, and tears the philosopher's stone out of his body, crushing it and effectively committing suicide.