Revealing mistake: With his unexplained amazing pyrotechnic and telekinetic powers Vigo lights up the Ghostbusters' dark room and traps them locking the door. Winston barges in and luckily his very mundane fire extinguisher easily puts down the hellish supernatural flames; it's so effective that it also extinguishes the flames in the foreground without him ever pointing the thing at them. (00:57:30)
Revealing mistake: Janine is wrapping up work at the phone and puts a cover on the computer screen. The monitor is plugged in, but the desktop is not. (00:55:20)
Revealing mistake: During the sequence where the ghosts knock the courtroom chairs in the air, you can see that some chairs have one leg that looks oddly 'loose' or with something sticking out (as interviews with the technical cast members clarify, each chair had an extension part that went into the floor and was launched up by an air mortar). (00:34:45)
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.
Revealing mistake: A Ghost Runner scares the people having a jog in the park. When you first see this peculiar ghost (who even checks his own pulse, funnily enough) the extras react to him with timing wildly off; the ones further back in the curve react when the ghost has already passed them by, but turn towards the middle of the track as if he were right there. One of the last people reacting is a woman who didn't even make it to the curve in time to see anything. (00:37:20)
Continuity mistake: The first shot of Lady liberty walking down the streets and looking down at people is an obvious human face in disguise and makeup. In the close-up shot, it is a mask/miniature that doesn't look like the previous one, but does resemble the real Lady Liberty. This is not due to the statue being animated by ghosts, for the mistake takes place between one shot and another (besides, the statue is animated all the time), but because the FX was achieved combining a human in disguise and a miniature set. (01:29:50)
Continuity mistake: When Dana is watching the couple arguing she has her arms crossed. A frame later, from a different angle, they are not.
Revealing mistake: When the slime jar bursts into flames because of the judge's malevolence, you can easily see (and from multiple camera angles too) that the flame does not erupt from the jar itself but rather by it. (00:32:30)
Suggested correction: The slime is overflowing from the jar, the explosion is coming from the slime itself, whether it is slime that is still in the jar or slime that has spilled onto the table.
Other mistake: Ray obtained a small sample of slime through a very little device, a sphere smaller than his fist. The slime is brought back up in court as "exhibit F" and it is said to come directly from the trunk of their car; the sample is almost 1400 ml, way larger than it should be. (00:25:20 - 00:29:05)
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.