Big Lipped Alligator Moment: A cop on a motorcycle drives into the TARDIS and comes out moments later to go god-knows-where while the Doctor and Grace look on.
Fandom Berserk Button: The movie introduces the notion that the Doctor is half-human on his mother's side. This is not something the fandom has embraced with open arms, and you'd be wise to not bring it up in discussion with them.
Fanon Discontinuity: Whether the movie is considered canon or not depends on which particular fan you talk to. However, almost everyone agrees that at the very least, the Doctor never said he was half-human (or if he did, then he was lying).
The Doctor, upon regenerating inside a morgue—specifically, in a cold chamber, while an orderly is outside watching Frankenstein. Somehow, the new, very confused Eighth Doctor kicks down the morgue door, attracting the attention of the orderly, who walks in, sees Paul McGann walk out wrapped in a sheet squinting at the heavenly light surrounding him. The orderly takes the only sensible recourse, and drops in a dead faint.
The Master's device he plans to use to steal the Doctor's body looks suspiciously like a Chameleon Arch.
Paul McGann wore a wig in the film because he had short cropped hair at the time that was deemed inappropriate for the Doctor. His successor Christopher Eccleston had the same hairstyle and didn't have to wear a wig.