As the "fake police car" leaves the starting point in NYC, it gets hit by the a undercover police car on the rear left side of the bumper and trunk, but in the next shot the "fake police car" shows no signs of damage.
During the scene where the racers are walking the streets of NYC to the restaurant, there is a taxi in the foreground with the time/temp reflected on its rear passenger's window. The time shows (mirrored) 6:45. However, seconds later, in the same tracking shot, Roscoe says, "It's 9:05, I'll will check in again at 10".
Toward the end (at about 1:28:50 we see Lapchik (the motorcyclist) on the highway. His jacket is almost totally gone; he's naked from the waist up other than a few tatters from his T-shirt. A short while later, in LA (starting at about 1:29:30, then at the very end, at 1:43:37, when he crashes into the ocean) he still has his full jacket.
All of the set-piece scenes with Roscoe and various state Highway Patrol units trying to bust the ralliers were shot on the same location, so that mountains are visible in the background in sequences supposedly taking place in flat Midwestern states.
While being pelted with bugs, Lapchick lifts his right hand off the throttle to wipe his visor. He should have immediately slowed and the tone of his engine should have changed. Proof that the motorcycle and Lapchick were being towed on a trailer while being filmed.
Some motorcycles of the time have throttle lock switches, that would enable the driver to lock the throttle in a desired position. If Lapchik's motorcycle was equipped with this feature, this would in fact, be possible.
When the Acme Plumbing van is running from the trail of burning gasoline, you can see the fire extending in front of the van. It is most noticeable right after the passenger says "You can't out run it".