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.
Die for Our Ship: A large number of the fandom actually supports Betelgeuse/Lydia. This has a lot to do with people watching the TV series when they were kids, which portrayed The B Man as having strong feelings for Lydia bordering on romantic. It's slightly worrying - quite apart from Beetlejuice having been dead at least a century, Lydia is disgusted by him in the movie. But more notably she's only 12-14 in the series, where in the movie she was 15-17.
"Funny Aneurysm" Moment: Sylvia Sidney plays a smoking ghost with a hole in her throat which emits smoke. Sidney died from throat cancer in New York City - at the age of 88.
Hilarious in Hindsight: The original concept of Beetlejuice being a shape-shifting reptilian demon. One of the insults used against Ghost of True Capitalist is that he's a reptilian shape-shifter.
I Am Not Shazam: Due to their being homonyms, people make the mistake of thinking the character is named Beetlejuice as well, rather than Betelgeuse.
His name is officially "Beetlejuice" in the animated series - hence Lydia calling him "BJ."
The Maitlands have difficulty summoning him at first, as Adam thinks it's pronounced "Bay-tell-gice."
Additionally, despite the poster, the film credits make it clear that the title is Beetle Juice, not Beetlejuice.
In the debut episode of the cartoon, Lydia was going to kiss B.J., until she saw the bugs in his teeth.
Depending on your preferred level of Squick, this COULD be either The Jailbait Wait (low squick) OR Wife Husbandry (high squick). It's only six years 'til Lydia becomes an adult, after all, and Beej has all of eternity to wait.
Hilarious in Hindsight: When Claire Brewster loses election as class president in the episode "Running Scared," she immediately demands a recount.
In Name Only: Due to the different premise, Lydia and Beetlejuice are friends now, and the Maitlands and Juno are nonexistent. Lydia is also several years younger than the movie version.
Squick: You think Beetlejuice's crush on Lydia was creepy? At least BJ is relatively human - in one episode of the cartoon, Lydia gets the unwanted attentions of an anthropomorphic bull ghost who wants to marry her. Beetlejuice even sees what Lydia's future will be like if he doesn't go back to rescue her... Yikes.
If it wasn't bad enough, in his nightmare there's little baby heifers running about all over the place. Ew...