These are what we call the 'YMMV items.' Things that some people find in this work. We call them 'your mileage might vary' because not everyone sees these things in the same way. This starts discussions in the trope lists, a thing we don't want. Please use the discussion page if you'd like to discuss any of these items.
Accidental Aesop: The townsfolk repeatedly call Belle "strange" and "odd", presumably because she is a bookworm and ignores the local jerk. So the lesson of the song is, "If you are kind, chaste, and intellectual, people will think you are weird and your only friends will be talking furniture." This could, however, also be taken in the in-verse, seeing as Belle is quite clearly the protagonist and the folks are not all portrayed positively. "Remain true to yourself despite what others think."
Alternate Aesop Interpretation: Some people have argued that the Beast's story is a metaphor for the AIDS crisis of the gay community in the 1980's, when the movie was made (there's even Reality Subtext here, as composer Howard Ashman died of AIDS just before the film was released). The argument goes as follows: the Beast is hit with a debilitating curse that strips him of his humanity; he's forced to live in complete isolation, hating the world that he was once a part of; the villagers, upon hearing about the Beast, give into threats that he'll harm their children and generally express their hatred of anything unlike themselves ("we don't like what we don't understand, in fact it scares us, and this monster is mysterious at least..."), with the (straight) men of the town deciding to brutally attack and kill the "different" person; and they're led by an extremely macho villain who is wholly motivated by his heterosexuality and spreads baseless lies about the Beast for his own gain. This is even more apparent in the Broadway version, where it's made clear that the enchantress's spell works like an illness that gradually makes the Beast grow worse.
Perhaps the biggest one (that even inspired a lot of debate and even fights behind the scenes) is who is the actual protagonist? Belle is certainly the viewpoint character, and most of the story is told from her perspective, but the character who develops and changes the most as a person (and whose actions drive most of the plot) is the Beast. Howard Ashman in particular was convinced of this and modern critics are more and more inclined to agree.
In a departure from the original story in which the Beast was kind and gentlemanly (if very rarely hot-tempered), the Disney character starts off angry and depressed and has all the classic trappings of the villain.
Commentary from the filmmakers has fueled theories that the Beast is near-suicidal for much of the film and his rescue of Belle from the wolves as also an attempt to kill himself while doing something meaningful.
Belle: Is she a well-meaning bookworm unfairly made an outcast by the villagers or is she actually a snob, and therefore deserving of being ostracised by the village? Is her falling in love with The Beast genuine or the result of serious Stockholm Syndrome towards an abusive captor? Trying to puzzle this out has led to many an Edit War on the Family-Unfriendly Aesop page.
The song Be Our Guest. Do the servants want to make Belle feel welcomed in the castle and be kind to her, or are they only being extremely courteous to her at dinner because she is able to change them back into humans?
The villagers. Are they misguided people who are genuinely afraid that the Beast might harm their children? Or are they true monsters of this film, since they're the ones who put Gaston on a pedestal and back his more devious plans? The fact that they willingly supported Gaston's plan to blackmail Belle into marrying him, despite his revealing enough of it to know how horrific of a plan it is, with absolutely no sign of fear, strongly supports that theory.
Then there's Gaston. Is he a complete jerk? Is he actually a fun guy with a zest for life who thinks that Belle has been driven mad, because she seems to think that this monster is a nice guy? Is it him who winds up going mad after Belle rejects him and humiliates him in front of the entire town? It's worth noting that he was originally supposed to die by falling off a cliff and laughing hysterically, indicating that he had indeed been driven mad in his desperate effort to impress Belle.
"Something There", though not as famous as the others on this list, is what a song in a musical is supposed to do; it encapsulated a massive leap in character development that would have been incredibly awkward no matter what dialogue you tried to use.
Draco in Leather Pants: Gaston has a large rather large fanbase who are willing to overlook the fact that he is a misogynistic, arrogant, murderous Yandere, some to the point that they think Belle is either crazy for rejecting him, or a stupid bitch that he didn't deserve anyway. It doesn't help that he's been played by Hugh Jackman, among others.
Lumiere is pretty popular, mainly thanks to his charming personality and performance of "Be Our Guest".
Lefou and the Wardrobe are very minor characters, but they're so much fun that almost everyone remembers them.
Babette, the feather duster, is also a fun character and gets a much larger role in the stage version. In almost any production, about half of the girls whom you'd think would be auditioning for Belle actually want to be Babette.
Monsieur D'Arque. He's only in the film briefly, but Tony Jay gives such a memorable performance that it's easy to remember him.
Evil Is Sexy: Gaston. He's devilishly handsome and everyone knows it, but he has no inner goods....
Heartwarming In Hindsight: Gaston tosses away Belle's book and tells her that it isn't right for women to read or even think. When Belle finishes reading a book to Beast, he immediately asks her to read it again.
Hilarious in Hindsight: Elisa and Goliath of Gargoyles once spent Halloween as Belle and the Beast. Gargoyles' equivalent of kissing is stroking their lover's hair. Think of this during the 3 times when the Beast runs his fingers through Belle's hair.
About the part of the castle where the rose is located. It's The West Wing. Really. Repeated mentions of the West Wing being forbidden just add to this.
Ho Yay: LeFou is slavishly devoted to Gaston; some people also read a bit of subtext into Lumière's interaction with Cogsworth (they certainly do have a bit of Tsundere-esque bickering going on).
Hype Backlash: This movie's status as the only traditionally animated feature to earn a Best Picture Oscar nomination has led some people to call it Disney's most overrated film.
Love to Hate: Gaston. In other mediums too—throughout the show's Broadway run, whenever the actor playing him came out for the curtain call, he was heartily booed by the audience, much to the man's amusement/enjoyment.
The beast himself is meant to be a horrifying monster, but once the audience gets used to his appearance, most consider him to have a fascinating design and prefer it to his human form.
Moral Event Horizon: Gaston starts out as pretty harmless. An arrogant jackass, but harmless. Then he plots to throw Maurice in the loony bin unless Belle agrees to marry him and plots to kill the Beast because Belle prefers Beast over him, but it's when he stabs Beast in the back after Beast spared his life that you know he's beyond redemption.
Nightmare Fuel: In the Broadway production, the servants weren't straightforwardly turned into wacky talking objects. They're cursed to live as hybrid-object-human things that are slowly turning into normal inanimate objects. One wonders whether they'd still be conscious when they've fully transformed or just dead - either is pretty disturbing. Cogsworth and Lumière (themselves transforming into a clock and a candelabrum respectively) even joke about one man who has turned into a brick wall. Of course, it's really a concession to the fact that the parts have to be played by people - so the scene explains why the clock, candlestick, teapot and others are still "human sized", but it still makes for excellent Nightmare Fuel.
One-Scene Wonder: Monsieur D'Arque (the guy who runs the insane asylum). His voice actor, Tony Jay, did so well with the small-but-villianous role, that Disney decided to give him a much bigger role as the primary antagonist in The Hunchback of Notre Dame.
They Changed It, Now It Sucks: The Remastered version is this. Whatever happened to all the shadows and gloomy colourings? The whole point of Belle's "come into the light" line was originally so we could see the Beast in all his grizzly glory for the first time since before he was no more than a creepy face in the shadows - we see him clearly as a Beast in his first scene with Maurice in the Remastered version!
So, Belle, you're being held prisoner by a giant monster with serious temper problems who is starving you because you wouldn't eat dinner with him. So what do you do? You go into the one place in the castle he specifically told you was forbidden. And then uncover and try to touch a glowing, floating rose.
Also, Belle having to use the mirror to expose the Beast's existence, of all things, to keep her father from going to the Asylum. Really, Belle, did you honestly expect Gaston or the rest of the villagers, who were congregated into a lynch mob at the time, to even think that the Beast was friendly, and not try to kill him?
Older Than They Think: Many people will be surprised to say that much of the elements found in Disney's version originally came from this. The animate furniture, the fact that Beauty's name is Belle, the magic mirror, hell, the character of Gaston was heavily based on a character in this movie who pretty much served the same purpose.
Unnecessary Makeover: Many viewers were disappointed with the fascinating Beast's transformation into a generic prince, with Greta Garbo famously saying "Give me back my Beast!" as she left the theater. According to Cocteau, this was intentional.
Awesome Music: Sumptuous work by Lee Holdridge and Don Davis (nominated for five Emmys, and won three of them).
Seasonal Rot: Season Three, in which Catherine dies, is deeply unpopular with a lot of the fanbase.
Alternate Aesop Interpretation: This story has long been interpreted by some feminists as an object lesson meant to teach young girls the virtues of accepting an arranged marriage with a strange and scary older man (whose wealth would benefit the girl's family). But it could also just as easily be a fantasy aimed at wealthy and socially awkward men who fear that their wives will only love them for their money and position, not themselves. It's clear that Beauty's sisters only care for their own selfish whims, while Beauty herself is kind and dutiful, and willing to sacrifice her own life and freedom to save her father. Beauty is also willing to overlook the Beast's appearance and love him for his personality, not his resources. The Beast's transformation into a handsome form at the end could symbolize the return of his self-confidence once he managed to wisely pursue and earn the love of a high-value woman.
Ending Fatigue: Gabrielle-Suzanne de Villeneuve's version goes on for pages and pages after the curse is broken.
It Was His Sled: The fact that the Beast and the Prince are one and the same is supposed to be a surprise in the original story. Pretty much every adaptation tells you upfront right at the start.
Values Dissonance: In many versions the prince is cursed because he refused to give hospitality. Given that refusing people shelter in bad weather could be a Matter of Life and Death — it's not as if they could hop in their car and drive to a hotel — it was taken very seriously, so the curse definitely would not have been seen as Disproportionate Retribution.