Deliberate mistake: When the kids are about to jump start the trap Phoebe hides behind the school bus door. Half a second later, when the trap opens, she's nowhere to be seen, not even jumping or stepping inside the bus. Then half a second later she has managed to get inside, turn around and protect her head while the bus windows burst, and yet a second after the havoc Podcast and her are already outside, watching the ghost fly away. It all happens in real time for their hairs are still fluttering. It sure paces up the scene, but it's really awkward.
Plot hole: The whole premise of the movie is that history would write off the existence of the Ghostbusters after the events of the first movie. In that movie there was prolonged large scale destruction in the heart of a city with millions of inhabitants. It's simply impossible that people would forget or dismiss it. And that's if we do not even begin to assume that the second one happened, even if the director said it did; nothing in his movie shows that, and for a good reason (Statue of Liberty, anyone?).
Other mistake: When Gary is about to test the trap check the bus door on the right: there's a reflection of what seems to be his shirt, but it's a long sleeved wrinkled shirt, not the short sleeved he's wearing. What's even more bizarre: when Gary moves around, the reflection stays completely still and, drumroll, a shot later, from the very same angle, the reflection disappears. Either it was a crew member hiding, or a CGI-made reflection placed to hide some unwanted crew or equipment reflection.
Visible crew/equipment: When Gary opens the trap next to the 2 school buses, he is wearing a cable attached to an earpiece on the right side of his head. The kind of cable actors wear to receive instructions from the crew. There's no reason plot-wise for him to be wearing it (he isn't receiving any instructions from the kids, who are right next to him).
Continuity mistake: Callie finds the 'secret' (even if it's in plain sight, really) basement with Egon's stuff following the instrument she found in Phoebe's room. She has the tool with her, she does not have it with her when she is possessed, but it is back at the house when everyone else is home.