Character mistake: There are signs out front as the Spenglers arrive at the dirt farm. One of the signs is a call back to a scene in Ghostbusters 1984. In the scene, Ray quotes the book of Revelation: "And I looked, and he opened the sixth seal, and lo, there was a great earthquake. And the sun turned black as sack cloth, and the moon turned blood." When Finn Wolfhard reads it he misquotes the line saying "The sun became as black as cloth."
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. It all happens in real time for their hairs are still fluttering. It sure paces up the scene, but it's really awkward.
Audio problem: During the Walmart scene, it seems they edited in Gary saying "Oh, God." However, it's over no visible mouth or throat movements. (01:15:17)
Continuity mistake: When they are chasing the 'muncher' ghost down the street, the light bar is only seen flashing on the Ecto 1 when seen from inside the restaurant. The rest of the time they are not on. (00:59:00 - 00:59:30)
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).
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: In Walmart, some mini-Stay Puft Marshmallow Men turn on a propane grill and set each other on fire. Gas grills on display never, under any circumstances, have propane tanks installed. This would create an obvious safety issue because there is nothing keeping a customer from accidentally (or deliberately) starting the grill and causing a fire. Additionally, propane tanks are never stored inside for similar safety reasons.