Erik, the titular character in The Phantom of the Opera is a disfigured, tortured soul, longing for compassion from another human being...who also happens to be obsessively stalking a girl and in serious need of therapy. He kills at least two people, sabotages a chandelier—which could have injured or killed even more people (the TV miniseries depicts exactly this in the aftermath), kidnaps Christine (and quite possibly takes advantage of her while she's entranced/unconscious), blackmails her by threatening to kill her fiancee unless she marries him, and essentially forces a world-renowned opera house to put on his self-insert fanfiction - which he then literally inserts himself into! A lot of fans tend to give him the Woobie treatment and ignore his terrible actions, as well as romanticize his relationship with Christine and portray Raoul as a jerkwho doesn't respect her.
The fact that the musical itself romanticizes the Phantom certainly helps. Gaston Leroux's original portrayal in the novel was much harsher and less sympathetic. And of course, Andrew Lloyd Webber noticed many fans were unsatisfied with the ending, so he decided to create a little sequel called Love Never Dies; fanning the flames of Leather-Pantsing and making all militant Erik/Christine shippers' dreams come true, and arguably invalidating the point of the original musical.
The fact that so many actors playing a character defined by his horrifying disfiguration manage to bring some serious sex appeal probably contributes.
Phantom also has an in-universe example in the form of Erik's own opera, Don Juan Triumphant. From what little we see of it, it appears to be a rewrite of Mozart's Don Giovanni, with the title character avoiding his Karmic Death from the source and getting the girl in the end. This, of course, fits perfectly with Erik's character given that he sees himself as a misunderstood genius and lover, so of course he would identify with Giovanni rather than his victims.
A Streetcar Named Desire. Stanley Kowalski. People tend to blame Blanche for being a passive-agressive weirdo and kind of leading him on, but that doesn't change the fact that Stan is an evil bully and a rapist. Within the story, Stella refuses to believe that the rape actually happened. What really hurts Blanche's case is establishing herself as a bully and a racist in one line to Stella ("I let the place go? I let the place go?! Where were you?! In bed with your Polack!") before Stanley first appears. This combined with Stanley countering the same epithet later by saying he's a full-born American right before his assault surely makes a lot of people want to justify his actions, even though they can't.
Deliberately invoked in many productions of A Doll's House, where the actor playing Torvald Helmer has to be ridiculously attractive for Nora's actions to make sense. Unfortunately, it's easy to miss that he's also an ungrateful sexist bastard.
Nils Krogstand, the Designated Villain of the first two acts, easily is subjected to this. Though given Ibsen uses the first bit of the third act to practically turn him into The Woobie (depending on how the actors play the infodump of his backstory with Mrs. Linde) it might have been a deliberate subversion of the stereotypical theatrical villain of the time.
Assassins is prone to causing this, especially depending on the attractiveness of the cast. Booth probably gets the most, due to being a "handsome devil".
There is a significant population who woobiefy either or both of the Macbeths, which is usually done in conjunction with this.
Both Iago and Edmund receive a little of this treatment. The fact that they're both intensely charismatic (part of being the Magnificent / Manipulative Bastards) and clever young men helps, as does their ambiguous motives (Edmund's given complaint being a logical fallacy), but ultimately the clencher is how good-looking the actor in question is. Since the former has been played by Kenneth Branagh and the latter by Philip Winchester... yeah...
The Merchant of Venice: Shylock is what happens when this attitude drifts down into academic circles, and then into the general public. Nearly all modern interpretations transform him from the (admittedly very unsettling) antagonist to a tragic character, while the intended heroes are racist assholes bent on ruining his life. Nevermind that Shylock's entire scheme was to murder Antonioover him insulting Shylock's profession. And then his daughter (who also gets this treatment, despite being little more than a greedy bitch according to the play) runs off with most of his money and his family heirloom. Modern audiences usually treat both interpretations as being antisemitic despite those traits being character traits of Shylock and not a commentary on the Jewish people as a whole. It's easy to forget that, prior to about the 19th century, anti-Semitism was based not on racism but on religious bigotry (indeed, most Europeans believed that they, themselves, were descended from the Jewish people before the Indo-Aryan hypothesis was put forward by German linguists), and in the play Jessica is a "good Jew" because she wants to become a Christian. Apparently, since Shylock himself is forced to convert at the end of the play, the implication is that either he'll undergo a Heel–Face Turn or he'll just continue to be evil - this time, as an evil Christian.
People also tend to Leather Pants Richard III in Shakepeare's play of the same name. True, he wasn't nearly as bad in real life and this play is one of the most famous cases of Historical Villain Upgrade, but he's still an evil bastard in the play. This is probably a case of Dated History, as it was long believed that Richard III did commit many of the atrocities he did in the play. This tendency may go back to Shakespeare's own lifetime, given this story.... It helps that most of the characters in Richard III seem rather boring and stupid compared to the charismatic, witty Richard, who is continually talking to the audience and cracking jokes about how evil he is. Even the supposed hero, Henry VII, only comes in at the end so doesn't come across as an easy to root for character.
Journalist Caitlin Moran applied it to Judas. Though it is the musical version:
In Jesus Christ Superstar he gets played by a hot black dude in rad trousers, and his libretto poses a series of extremely perceptive theological questions, viz, the wisdom of Christ’s tactics in declaring himself the Son of God. And did I mention his trousers, already? They really are superlative trousers. Really ... you know. Phwoargh. Good trousers. Yeah. Judas comes out of Jesus Christ Superstar really well. Had they made “Team Iscariot” T-shirts in 1973 I would probably have bought one.
Freddie Trumper of Chess is arguably sympathetic, and arguably a genius, but he is also a chauvnistic, manipulative, paranoid asshole who drives away the only person who can stand him by saying almost unbelievably cruel things about her presumed dead father when he sees her talking to his opponent. You would not know this from some of the fanfiction he gets. Granted, there is lots of room for interpretation of all the characters depending on which version you see, but an angel Freddie ain't.
Gabriel Goodman of Next To Normal may be a subversion, as he is portrayed in canon as extremely seductive/alluring in canon and is DIL Ped by Diana herself. He also gets several EXTREMELY sexy Villain Songs.
Gabe's case is also debatable in that he's not necessarily a villain; who he is depends on who he's interacting with. What Diana needs from him is not the same as what Dan does.
Aaron Tveit's taking full advantage of the set consisting mostly of metal poles probably didn't hurt.
Judge Turpin can get this in productions where the song "Johanna" is kept. This song gives him more depth, portraying him as a tortured person who knows his lust is wrong but is unable to keep it in check. However, many use this as an excuse to absolve him of everything
The same song could arguably make him even more repulsive, not least because it shows that this "unchecked lust" and "extreme action motivated by said lust" problem of his is a recurring and ingrained trait.
Death in Elisabeth intentionally invokes this. Lucheni, Elisabeth's assassin, can also get this treatment if he's played by the right actor.
Both of the von Krolocks, father and son, in Tanz der Vampire. Literally, in a few costume sets.
The fandom applies this treatment to the show's version of vampires in general, including continuing to depict Alfred as The Ingenue in fic/art set after the end of the story, despite it being made very clear in the final scene that he and Sarah have become as evil and heartless as any of the other vampires upon being turned and are more or less the catalysts for vampire world domination.
John Dickinson has a lot of fans despite campaigning vehemently against American independence seemingly on the grounds of protecting rich men's interests, although he proves at the end that he is just as principled as Adams and leaves Congress to protect the country as a soldier after failing to protect it politically. (And the characterization was a Historical Villain Upgrade in any case.)
Edward Rutledge, the man who forces the deletion of the anti-slavery clause from the Declaration of Independence,note show only, not historically is also quite admir'd by some for his elegant Southern Gentlemanly manner. He was played by John Cullum in the film, though, and his astounding voice is very effective in "Molasses to Rum."
Draco mocks everyone (lamely), but it turns out that he's madly in love with Hermione, and he ends up teaming up with everyone at the end to kick out the Death Eaters and bring down Voldemort. It's even more extreme in the sequel, in which it's revealed that Draco just wants to be loved by his dad and become Harry Potter's best friend.
Voldemort himself. A lonely Woobie who just wants companionship and actually does get redeemed at the end. It helps that fans adore his Ho Yay filled (and later canon) relationship with Quirrel, and his manyfanservicescenes.