In the case of the Phantom, it's done on purpose in the Broadway production. However, the original novel is much more straightforward about how we're supposed to interpret that character.
On the other hand there is a lot of this concerning Raoul and especially Christine. She has been interpreted as a child-like idiot savant, a young woman suffering from a severe Electra complex, and even a straight-out victim of sexual abuse.
Big-Lipped Alligator Moment: In the original novel, a character known as the "brown man" is introduced, living in the sewers parallel to, and only intersecting momentarily, Erik. It's explained that he's a hermit monk, and that he's just always been there. Erik is understandably more frightened of HIM than he is of Erik. He is never mentioned again. Interestingly, this character is a lot closer to the real figure Erik was loosely based on.
Die for Our Ship: Raoul, more often than not, is done away with so Christine can hook up with Erik.
Draco in Leather Pants: Erik's tragic backstory and twisted love for Christine is used by certain segments of the fandom to overlook and excuse all of his evil deeds, such as kidnapping and murder.
The Persian and Erik — Raoul doesn't help by comparing the ways the Persian and the smitten Christine swoon over the charismatic Tragic Monster
Occasionally some between Erik and Raoul, of all people. Mainly due to the fact that Erik apparently keeps very close tabs on Raoul, even to the point where he breaks into Raoul's house to watch him sleep...
Macekre: The English translation by Alexander Teixeira de Mattos, which was the only one available until 1990. It's now in the public domain, so any English edition that does not specifically credit a translator by name is most likely de Mattos' Macekred version.
Narm: Due to the heavy use of Purple Prose within the novel it was inevitable that we'd wind up running into this trope. A notable example is during the climax when Erik pulls a And Now You Must Marry Me. What is set up to become a tense scene becomes unintentionally funny when he announces "and the grasshopper hops jolly high!"
The Scrappy: Raoul might be one of the least popular heroes in all of literature, being jealous and possessive of Christine even before they're actually dating, not being particularly intelligent, and even berating her on occasion.
Vindicated by History: The original book isn't much more than a frothy pulp horror novel and hardly left an impression on the general public upon its publication. It wouldn't be until the many, many adaptations it received that it would be considered any kind of special.
The famous musical and myriad other adaptations further contain examples of:
Ramin Karimloo and Sierra Boggess, who played respectively the Phantom and Christine in the original Love Never Dies and the 25th anniversary at Royal Albert Hall, get a lot of this from the phandom, due to their chemistry onstage and in real life, with the term "Rierra" designating them. There are frequent arguments about them dating or not... until one discovers Ramin is actually happily married with two kids.
Gerard Butler and Emmy Rossum, or "Gemmy" still have a devoted fanbase that has been shipping them since the movie came out.
Adaptation Displacement: The popularity of the musical has eclipsed the previous works in the franchise, including that it originated as a novel. Many know there are movies but quite a few assume they're adaptations of the play (which is mistaken for far older than 1986).
Alternative Character Interpretation: Erik has become more or less the poster-child for woobiedom. Try to find a fic where he ends up miserable and alone. The fan base is split over this, but a strong majority finds Erik to be by far the most sympathetic character in the book/play/film. And how you feel about this will make a difference on what you think of Love Never Dies. Fan works aside, there's been a lot of different interpretations of who Erik actually is—he's been portrayed as everything from a doomed romantic who just got pushed a little too far (Kopit/Yeston version) to a slasher-style killer who sold his soul to the devil and flays the skin from his victims (the 1989 film starring Robert Englund, yes, Robert Englund).
Christine. Is she genuinely in love with Raoul, or is she unconsciously attracted to him simply because he can save her from her Stalker with a Crush? Notice on the rooftop scene, after Raoul declares his love for her, Christine immediately says, 'Order your fine horses, be with them at the door...'
When she kisses The Phantom at the end, is it because she really loves and/or pities him or because she's trying to save Raoul?
It's interesting to note in the stage production that while both Raoul and The Phantom explicitly say they love Christine and make that declaration to her, she never says it back to either of them, not even in song.
Big-Lipped Alligator Moment: The "mirror bride" bit is just bizarre. Yes, it provides a great romantic moment for the Phantom to catch the fainting Christine, but makes little sense. (Although given that he later forces Christine to wear the same wedding dress, he's presumably showing her his plans/wishes for her).
Most of the comments on YouTube clips of the newly designed US tour are not kind, decrying the many changes made. But there's a decent amount of praise, and many people agree that they actually prefer the tour's version of the opening sequence and chandelier crash.
Critical Dissonance: When the show opened, reviews were not flattering. But in October 2016, the show celebrated its 30th anniversary in London and it reached the same milestone on Broadway in January 2018.
Fan-Preferred Couple: The vast, vast majority of The Phantom of the Opera fanfics have Christine ending up with Erik.
A lot of fans would like to forget The Phantom of Manhattan, a novel by Frederick Forsyth that was based on the original plans for a sequel to the musical in the late 1990s, ever happened. Love Never Dies appears to be heading in the same direction.
Also, the canon ending of the original story, where Christine ends up with Raoul. Try to count the number of fanfics that have her ending up with Erik instead. We'll wait.
Especially so upon watching the guest performance at the end of the Royal Albert Hall performance. She was actively inching away from him and grabbing Michael Crawford's hand. (ALW managed to hold her other hand, though.) Although Sarah's still smiling and laughing, she looks like she wants to get as far from ALW as humanly possible.
Considering the ephebophilic overtones of the story, it's a bit cringe-worthy to know that James Barbour, Phantom on Broadway from February 9, 2015 until December 23, 2017, was convicted of sexually abusing a fifteen year old girl in 2006. Despite vociferous protests from the theatre community, he retained his post through multiple cast changes.
Hilarious in Hindsight: The 1937 Chinese version, Song at Midnight, features a Phantom who is disfigured in an attack rather than born deformed; while its very unlikely that Western film producers were aware of it upon its release (as it didnt receive a release outside of China until the 1990s), many subsequent versions just happened to include a similar divergence from the book.
Some Phantom fans (or "phans") say that the makeup on Phantom is a cop-out, as it's still only over one half of his face, but, let's be honest - a man who's good-looking on one side and like a rotting steak with an eye on the other alongside a partially exposed cranium and about as much hair as the average healthy person cleans out of their brush at the end of the month is still deformed enough to be believably outcast from Victorian society, even if he's not the "living corpse" Leroux described. (And with those conditions, being handsome on the other side tends to only make the deformity look worse.) Word of God says that:
The half-mask was created because it was very difficult for Michael Crawford to sing properly through a full-face mask,
Both Crawford and Andrew Lloyd Webber have said that it just took too long to apply make-up all over his face; Lloyd Webber was apparently worried no actor would play the role for very long if it required five-plus hours in make-up. Even with a half-mask, it takes time: by the end of Crawford's tenure, the make-up application was down to two hours, and that's about how long it takes for his successors.
And, in addition to all that, it's very hard for an actor to emote to people at the back of the theatre when most of his face is covered half the time.
And in the Takarazuka Revue productions of the Arthur Kopit/Maury Yeston musical, both Wao Youka and Haruno Surime◊ were WAY too pretty for their own good. Bizarrely, they both managed to make it work regardless.
Magnificent Bastard: The Phantom, born Erik, spent his childhood as a freak in the circus on account of his disfigured face. Helped by Madame Giry to escape, the Phantom sets up residence beneath the Paris Opera House, styling himself as a ghost to "haunt" the venue and have it perform to his expectations. A genius singer, composer and illusionist, the Phantom took Christine Diae and taught her how to sing so that she would glorify him on the world stage and eventually fell in love with her. When the Opera House's new owners attempt to defy the Phantom, he sabotages and murders to get his way, boldly presenting himself at the Masquerade and forcing the House to perform his masterpiece, Don Juan Triumphant. An attempt is made to catch him, but fails, the Phantom absconding with Christine, only releasing her when her compassion makes him realise his cruelty, letting her go so she can be happy. In the sequel the Phantom lures Christine back to him to give her a chance at a comeback, wins her back from Raoul and almost has a completely happy ending if Christine hadn't tragically died. Phantom still manages to end the story reconciling with his son, giving hope for the future. A tragic artist, the Phantom has seduced audiences for decades with his dark charisma and romantic Music of the Night.
The conceit that only half of the Phantom's face is deformed, originally designed because the full mask was interfering with Michael Crawford's singing, has become the iconic visual for not only the Phantom himself but any similarly deformed character.
On Tumblr, calling Erik a "melodramatic sewer goblin" or just "sewer goblin," due to a text post calling him that.
The "mirror bride" bit. It's just. . . weird. It's been eliminated from the US tour and the 25th Anniversary concert, but this has resulted in further Narm—Christine's fainting is now abrupt and inexplicable.
Inverted. Raoul is widely hated in the fandom, but the number of people who hate him drops dramatically when the role is played by Hadley Fraser. "I hate Raoul, but I love Hadley too much to care!" is a common refrain, as is "Hadley is the only Raoul I've ever liked!"
Jordan Donica has received similar praise, much of which suggests he'd make an excellent Phantom as well.
Ron the Death Eater: Expect Raoul to get hit with this by the largely Erik/Christine shipping fanbase, who make Raoul out to be a horrible person to Christine (before Love Never Dies technically made it canon), despite the fact that Erik himself definitely has plenty of his own issues as well...
Take That, Scrappy!: The Phantom dishes out a handful of insults to Raoul every time he mentions him.
They Changed It, Now It Sucks!: As stated in the "base" post, many fans feelings about the redesigned US/UK touring productions. Mostly averted with the non-replica Hungarian/Finnish/Polish/Swedish/Norwegian/Czech productions, where praise is nearly unanimous—particularly keeping Christine a blonde as she was in the novel.
What an Idiot!: It still wasn't exactly smart of Christine to unmask Erik in the novel, but at least there she had the excuse that it was a full face mask. She thought he was trying to hide his identity from her and - quite rightly, all things considered - wanted to know who he was, justifiably assuming there was a perfectly normal face under there. Christine in the musical, on the other hand, whips off the mask of a man who keeps only half his face covered - and therefore is probably concealing it for a very good reason. Such as to hide a deformity, perhaps? And she's shocked that he starts screaming at her? (Though the 'little lying Delilah' and general rough handling was too much.) Oddly, the lyrics suggest she does think he's concealing his identity despite the fact it would make no sense. All of this is probably because originally, Erik was intended to have a full mask—the half-mask was a last-minute change for practical considerations, made after everything else was finished (this is also why the iconic poster has a full mask).
The current touring version switches the Idiot Ball from Christine to Erik. He removes his mask to bathe his face after loudly playing the organ, assuming that Christine will still remain asleep for a while longer. She walks up behind him, taps him on the shoulder, and gets a full view of his maskless face.
And in the miniseries too. Erik repeatedly warns her that his face is too horrifying to look at. She assures him that she loves him too much to be repulsed. He finally relents and takes his mask off. . .and it's so terrifying (though the audience never sees it) that she faints.
The Woobie: Erik gets this treatment in some versions, while in others he causes at least as much grief as he gets.
In addition to Crawford's celebrated Playing Against Type turn, the title character in the musical has been played by Paul Stanley (of KISS fame) and Robert Guillaume (until recently, the only black actor to play the Phantom. In all fairness, the casting department might be trying to avoid the Black Vikings trope, as well as the Unfortunate Implications of a black man lusting after and stalking a young white woman).
The 25th anniversary concert got a bit of this from the fanbase, as well — some were skeptical since the Phantom and Christine would be played by the stars of Love Never Dies (though, to be fair, they had previously played those roles in the original show as well); and since Raoul, Piangi, and Meg were played by actors who had no prior connection with Phantom while several popular alumni of the show were cast in minor background and chorus roles. Once the event actually took place, though, the casting was generally pretty well-received.
Gerard Butler in the movie, full stop. The man never had a singing lesson in his life, let alone sung professionally, prior to his casting and despite being a perfect tenor, those notes are just a bit out of his range (ironically, Webber claims that Butler's singing was the absolutely closest to the "rock star" voice that he'd always imagined the Phantom having).
To a lesser extent, the musical itself.
At long last, after 26 years, Broadway cast its first African-American Phantom (Guillame's turn was as part of the national touring company), Norm Lewis, who debuted in May 2014. Opinion seems to be that his acting and singing talents are transcending the problematic above-mentioned tropes, as they did during his turn as Javert in the 2006 Broadway revival and 25th Anniversary Concert of Les Misérables.
And the move has opened up opportunities for other minority actors, with Derrick Davis and Quentin Oliver Lee as now the third and fourth, respectively, African-American actors to play the role (in the touring company), Jordan Donica as Broadway's first African-American Raoul, and Ali Ewoldt as Broadway's first Asian Christine.