When Eric first goes back to his apartment, right after he returns from the grave, it is raining inside the apartment and his movements look an awful lot like the movements he just made in the rainy alleyway. [Not really a mistake, but worth noticing - the filmmakers were forced to re-use some shots after Brandon Lee's death.]
Right after Gideon's Pawn Shop blows up and the police officer and Eric are talking, the officer yells at the people looting the shop; one looter is running away with a TV set. This is actually James O'Barr, creator of The Crow comic series! He was on the set one day and the filmmakers asked if he wanted to be in the movie.
In one of the flashback scenes, we see T-Bird reading from a book - "Abashed the devil stood, and felt how awful goodness is." The book is Milton's Paradise Lost. (NOTE: the line after this sounds like it is also a quote from the book, but it is not) David Patrick Kelly, who played T-Bird was so into his character that he bought an antique copy of the book. It is this we see him holding in the scene.
The latter part of the scene with Eric and Albrecht in Albrecht's apartment was actually ad-libbed. Brandon decided to pick up a picture of Albrecht and his wife and ask him about it. The following part about Shelley ("Believe me, nothing is trivial") was also not in the script. The scene was originally scripted to take place in the street outside the police headquarters, but it was decided that it should be moved to Albrecht's home so that the audience could learn more about him.