Continuity mistake: Watch the scene where Chili and Harry are driving past the Napoleon billboard. We then see a shot of Harry with the billboard quite far behind him, but when we cut back to Chili, the billboard is quite close again.
Other mistake: When the guy is on the balcony, and gets shot over the rail, you can clearly see the top rail bend to let him flip over and fall. Then when he lands you can see what looks like the corner of a mat lift up and kick dirt up. The next shot shows the bad guys leaning on the very rail that just bent.
Continuity mistake: When John Travolta, Rene Russo, and DeVito meet at Martin's house, Rene and John arrive in their cars and park facing the wrong way (driver's doors toward the curb), which in Beverly Hills gets you a ticket, but when they emerge from Martin's house, Chili Palmer's van is turned around (and the BMW that Rene Russo was driving is gone) so that the automatic door scene can commence.
Visible crew/equipment: When Chili and Harry are riding down the street and Harry is telling about the script that he wants to buy and why he needs $500,000, you can see the camera and crews reflection in the windows behind them.
Continuity mistake: When Chilli goes to see Leo in the hotel, the door swings shut when Leo comes in AND when Chilli leaves. But, when Leo goes back outside to argue with Chilli, it doesn't swing shut.
Continuity mistake: After Chili is searched and interviewed by the DEA agents at the airport, he leaves the room carrying the bag containing the book he bought in the airport gift shop. When Chili walks into the parking garage and meets up with Bear, Chili is empty handed.