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.
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.
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.
Angelina Dalles — Black Butler
Black Butler features a number of tragic deaths, but Angelina Dalles's is one of the most painful. Angelina is one half of the serial killer team Jack the Ripper, but she wasn't always inclined to such brutality. At one point, she was a kind, caring person who loved her family, and she even became a doctor in hopes of curing her sister's severe asthma. However, after her entire family is killed and she loses the ability to bear children, she freaks out and starts murdering patients who request abortions.
Angelina teams up with the Grim Reaper Grell Sutcliff, and the two of them kill with abandon. Eventually, Grell tries to push things further than Angelina is willing to go, and when she refuses to kill her nephew who she loves like a son, Grell ends her life. Despite the severity of her crimes, it's hard not to feel for her when she dies for showing a loved one mercy.