Visible crew/equipment: When the chairs are flying through the air in the courtroom, if you look closely, you can see the string pulling one of them. (00:34:30)
Trivia: The original theatrical version ended with Slimer eating the camera, just like in the first Ghostbusters film. This has been removed for the video releases.
Suggested correction: This is a popular rumor, and some people claim to remember it, but there's no supporting evidence or footage to confirm this. In fact, after doing some research, I came across an article where a projectionist screened an original 1989 35mm print of the film to see if it did indeed end with this mythical Slimer scene... and the print contained no such scene, ending the exact same way as the home video releases. Sadly, this appears to simply be another instance of the dreaded "Mandela Effect," where people remember something that didn't really happen/wasn't real. Furthermore, if they actually had spent the time and money creating an effect like that, what purpose would there be to cutting it from home video releases? Effect like the ones in this film were harder and more laborious to make in the 1980's. They wouldn't just cut effects out for no reason.
Continuity mistake: When the Judge states his sentence, from Stantz's POV of the front of the table the pink goo in the jar starts bubbling with a tag hanging on its right side. From a wider angle of the front of the table not only is the tag not visible, but the ghost trap next to it is different (check the number of dials and looks of the box), plus the trap is now slightly skewed as to how it was positioned before. (00:31:45)
Suggested correction: I believe the tag is visible at all times, at least the string of it (the tag itself in the wider angle from the defence's table is hidden by the jar itself). Not sure about the mentioned box, you mean the ghost trap? My impression is that the different angle accounts for a perceived different placement, but it's just my perception.
Revealing mistake: People are running away from the movie theater, terrified by the ghosts. Are they really though? The girl at the head of the line (she moves towards the screen from right to left, almost blocking the path of another notable extra that waves around a popcorn bag) just can't keep herself from laughing. Same for a guy that tries to whip the animated fur into submission using his scarf (although one could argue in that case that it's such a weird predicament that a bystander could be amused). (01:19:30)
Suggested correction: Exactly, some people are like the Ghostbusters, they ain't 'fraid a no ghost, which is why they laugh.
Other mistake: Dana leaves her home in such a hurry that she does not even put anything on her baby in the middle of December. She shows up at Peter's home, he phones Ray who then rushes out of the door saying he's gonna check Dana's apartment. Peter tells Dana that Ray is going to check her apartment, but their phone convo ended way before Ray had taken that decision, and he has no contact with the others till morning when they meet at the museum.
Suggested correction: Dana easily could have grabbed something that was laying on a table or piece of furniture that was on her way as she was running out of her apartment to dress Oscar in, or a good Samaritan could have given something as she made her way to Venkman's place. And Ray tells Venkman on the phone that he is going to check Dana's apartment.
Character mistake: At the psychiatric ward, Ray says that Vigo is a XVII century tyrant; it's not a very fitting way to describe him, since he saw the XVII century for just a decade; as the book mentioned, he lived from 1505 to 1610 and was the scourge of Carpathia for the XVI century. (01:16:15)
Suggested correction: Not a mistake, Vigo was a tyrant in the 16th and 17th centuries, Ray chose to label him a 17th century tyrant because that's the century he died in. It would be like calling Osama bin Laden a 21st century terrorist, which is correct, even though he also committed terrorist acts in the 20th century.
Plot hole: The Ghostbusters go underground to search for the slime using a map. Even excusing their initial dumb decision to try to dig a hole in the middle of Manhattan when they could have taken that route to begin with, there's no excuse why after their acquittal they never went looking for that river of slime until Vigo burns their lab; they spent weeks studying small samples of the thing but never tried to reach again the source of the infestation to monitor it, which is absurd. (01:00:00)
Suggested correction: The Ghostbusters know from Ray's initial journey down the hole that the river of slime is in the old pneumatic transit system. They are busy re-opening their business and will eventually go back to the river, but that mission gained more urgency once they linked the river to Vigo. Also, they know that the river has been there for years if not decades, and therefore realise that returning to explore it can wait a few weeks.
Suggested correction: This would need more clarification. I couldn't see any string in the mentioned sequences - and for the record, I don't think strings were used, you can easily find interviews with the production team that mention the different trick at work here (they quite literally 'shot' the chairs up in the air).
Sammo ★