Deliberate mistake: At the start of the film when the commentators are explaining the tie situation between the 3 cars the green arrow to the car is out of alignment and misses the car, instead pointing at the track. Pixar Creative Director, Jay Ward, who is this film's team manager (as well as art coordinator) has stated that this was deliberately done as a joke, for fun. It's a little bit off, in the same way that Chick Hicks is a little bit off. This is the video with Jay Ward explaining: https://youtu.be/ZkJre-eC-zE?t=25.
Continuity mistake: When R.J. pulls the wagon to stop Verne, The ground hook and chain is next to the right front wheel. But while R.J. and Verne are talking there's no chain or hook to be seen until they start pulling the wagon from each other. (00:40:25 - 00:40:55)
Continuity mistake: This is after news has arrived to the elephant village that Ayutthaya is fighting back after being occupied by the Burmese for some time. The raiders arrive and are rounded up before falling into a trap. At first, the pit is almost square, with rounded edges. Cut to an aerial shot and the same pit is now a circle. (00:50:45)
Continuity mistake: When Barris is served in the diner, french fries are scattered over his plate. When seen again, the french fries are replaced with vegetable sides.
Continuity mistake: After Lugnut says "Pedal to the metal" and the Champ's mechanic says, "Come on, Cruise, come on", the scene cuts to Cruise and Wrangler on the race course. Wrangler is ahead of Cruise and Cruise is behind Wrangler, to his right. In the next shot, Cruise is suddenly directly behind Wrangler. (00:28:35)