Continuity mistake: When Wally is whistling to lure the dogs away from the greenhouse, Kevin Spacey steps on his fingers to stop him. In that shot, you see that Spacey has the gun by his side. But in the next shot when you see his face, he's already got the gun raised.
Other mistake: In the film, each of the cop cars have a unique number associated with their plate number. Dave and Wally steal car "2286." Braddock and the other officer get into car "5210." After everyone was shooting at Dave and Wally, they're seen driving off while Braddock, Eve, and Kirgo are all bunched up together at the intersection. You can see Braddock's car number is now "2286."
Revealing mistake: When in the squad car, as they almost hit a couple of guys carrying a wooden bookcase, you can see the wig on the guy in the squad car is much bigger than Gene Wilder's hair.
Revealing mistake: During the car chase you can see it's Gene Wilder and Richard Pryor's stunt doubles in the car.
Continuity mistake: After escaping from the police station, they go into a building and break a wooden door to come out of the other side. When they first break the door, the broken section is only inches away from Richard Pryor. Gene Wilder says to watch out for it. When Richard Pryor moves it, it's slightly further away than it was before.