Continuity mistake: Near the end, the Ghostbusters make the Statue of Liberty smash the window on the roof of the art museum before abseiling in. The window gets completely destroyed, but just after Vigo says 'Now we become one' we see the window again and it appears less damaged than before.
Revealing mistake: When they gather around to watch what happens when you put the slime in the toaster, you can see when the toaster starts popping that it has little pneumatic pistons attached to the bottom, making it jump. When the toaster gets picked up off the table you can view its power cord which is entirely too big - more likely air lines.
Revealing mistake: When the Statue of Liberty is about to smash the museum's glass dome, the shot outside is really bizarre: The Statue is standing on a park, with a pink cloud covering what is supposed to be the museum. Yet the lower part of the cloud plus the people around it, show that there is no building nor any foundations there: Just a superimposed pink cloud over an empty field.