The eponymous Beast from Disney's Beauty and the Beast. He starts out seeming like the monster he appears to be, but we soon learn the anguish he feels about this behavior and how it's prevented him from becoming normal again (he even has a song about it in the Broadway version). His interaction with Belle then steadily pushes him into the realm of a Jerk with a Heart of Gold.
The DVD commentary discusses it in great detail, citing that no matter how bad he is your sympathy must always be with the Beast in order for the movie to work, and thus a lot of emphasis is given the constant emotional pain he lives in.
The eponymous protagonist of Coraline starts off the movie being essentially a self-centered brat, although, arguably, she does have a Freudian Excuse in her distracted parents. She also goes through a lot of heart-wrenching crap: Having to deal with the Other Mother wanting to eat her and keep her for herself and wanting to sew buttons on her eyes as well as nearly losing her parents and losing the Other Father and Other Wybie.
The hyenas from The Lion King. Their hatred for the lions isn't such a surprise given how they were segregated by Mufasa and forced to live in a barren wasteland with barely any food. They support the main antagonist, Scar, who promises a better life for them (and also revenge against the ruling family who've caused them so much suffering). It's only later that they learn Scar is utterly inept as a ruler and is willing to sell them all out to save his own skin.
The Lion King II: Simba's Pride has Nuka. His father chooses an unrelated cub over him, he's portrayed as The Unfavourite to his mother, and seems to be the Butt Monkey of Zira's pride. While he does want power, he also wants to gain the favor and attention of Zira, who is more concerned with Kovu. In the end, he dies trying to prove himself to his mother, and fails at that. He spends his last breath apologizing to Zira for failing, his death prompting one of the only displays of love and affection that Zira probably ever showed him.
Tigress as well, at least in the second half of the movie.
Depending on your interpretation of him and his Freudian Excuse, Lord Shen of the second movie. Considered too weak and sickly to warrant his parents' attention, Shen was left to the care of the Soothsayer. He strove to earn his parents' approval and ended up discovering the explosive potential behind fireworks. When the Soothsayer foretold his defeat at the hands of a warrior of black and white if he continued down his dark path, Shen slayed the pandas, believing averting his fate would finally earn his parents affection. They instead banished him and Shen saw this as the final sign that his parents did not and would never love him. He spends the rest of the movie killing and destroying out of a desperate need to fill the void in his life, knowing full well there is nothing he can do to make himself happy but also knowing there is nothing he can do to justify his actions.
Up: Charles Muntz - rejected by the scientific community, eluded by a stupid bird for sixty years, all by himself, going completely insane...it's so tragic...Of course, murdering innocent people because he thought they would take his bird away makes him a bit of a monster.
Shrek: Prince Charming. After enduring grief and humiliation now that he no longer has the power his mama gave him, all he wanted was his own Happily Ever After... which he probably would have gotten had he not stayed a villain by choice.
Perfect Blue: Rumi. She ends up delusional and tries to kill Mima. Me-Mania too. Yes, he was a serial killer and attempted rapist... But that's because he was a deformed, barely-functioning schizophrenic who lived only for his perfect image of Mima. The scene where we see his apartment is an enormous Stalker Shrine and that all he wants in life is "the real Mima's" affection was genuinely sad.
The eponymous character of Megamind. On one hand, he's an Ineffectual Sympathetic Villain who regularly kidnaps Roxanne, threatens citizens with bodily harm and always tries to kill his nemesis, Metro Man. On the other hand, he lost his entire civilization to a black hole, he grew up in prison because the inmates took advantage of his genius and naivete, and was ostracized in school by the kids because, unlike Metro Man, he looks different.
Princess Atta from A Bug's Life. She's rather strict towards her own younger sister, Dot. Not to mention refusing to see Flik's inventions as a good thing to help the ant colony. But she is rather nervous about her problems with taking over as a Queen that you're amazed she doesn't go further.
The ant colony itself as a whole. Sure, they brought the Darkest Hour on themselves by ostracizing Flik and PT Flea's entire circus after they hear the truth about the "warriors". But do they deserve the treatment Hopper gave to them afterwards?
Dave from Penguins of Madagascar. Once living at the Central Park Zoo, he had a roomy habitat and all the fans. Then started a chain where penguins unintentionally stole his glory as the octopus was moved to zoos and aquariums around the world, where he was forced to live in small tanks.
Prince Hans from Frozen. See his backstory for details and motivations of his actions. (i.e. growing up in a loveless family).
Joaquin from The Book of Life. On one hand he's fairly smug, has an air of Entitled to Have You towards Maria, and is a bit of a sexist. On the other hand he also had his father killed, his mother walk out on him right afterward, seems to be a predominately good person whose negative traits came from peer pressure/trying to live up to his father, and whether he knows it or not he's a Hopeless Suitor to the girl he's loved most of his life. He also lost his best friend who was like a brother to him and is being pressured to marry his other best friend when he clearly doesn't want to since they're both grieving, especially since she said she was only marrying him to get him to stay.
King Haggard from The Last Unicorn. The man was captured every living unicorn in the world and plans to harm our protagonist (and his own son!), but one can't help but feel sorry for his motivation: he's never been happy in all his life, and the only thing that ever gave him joy was the sight of unicorns. Anyone dealing with depression can surely understand, when your only source of happiness is one specific thing, and how that can make one act obsessive.
Lampwick and the other Stupid Little Boys in Pinocchio are a bunch of punks who enjoy destroying property, fighting, smoking, and drinking, but their ultimate fate ( being transformed into donkeys and sold into slavery) is far beyond what they deserve.
Scamper from Igor qualifies is you think about it. He's bitter, grumpy, jaded, cynical and by far one of the most sarcastic CGI characters ever created but he's also suicidal and immortal which most likely means he'll outlive all his friends and can't do anything about it.
Abbot Cellach from The Secret of Kells. He spends most of the movie as a control freak obsessed with building a wall to keep out the Vikings, to the point that he seems to care about nothing else and alienates most of the other characters. Then the Vikings attack, and he witnesses the deaths of nearly everyone in the village. Worst of all, he's lead to believe he's responsible for the death of his young nephew (and only remaining relative). After this point, he is a broken man who spends decades mourning what happened.
The prologue comic gives a lot more insight into his character. We learn that his sister, his brother-in-law, and their ENTIRE village perished in a Viking attack 10 years before the film, and the Abbot just barely managed to save his baby nephew Brendan. After going through that, it makes sense that he would dedicate his life to ensuring Kells would be protected from such an attack... and it becomes even more tragic that his efforts backfire.
Funnybunny from Dot And The Bunny is a mild example. He spends most of the movie trying to convince Dot that he's a joey and in general acts like an irritable and eccentric brat but at the climax of the movie he breaks down in tears and tells Dot that he was orphaned by hunters and is only pretending to be a joey to find a new mother. Also, keep in mind that rabbits are invasive species to Australia which may or may not push him into Jerk Ass territory.