Deliberate mistake: When Old Biff and Young Biff are sitting in Biff's car, the car's rear-view mirror vanishes during all shots of Old and Young Biffs together. Presumably this made the split-screen effect easier to do.
Deliberate mistake: While Indy is fighting the tank commander, he looks up to see them heading toward a cliff. After the tank goes over, from the time his companions look over the cliff to the time when they walk away and Indy is sitting on the ground, you can see the wind blowing in their faces. But as Indy starts to get up, the wind blows his hat toward him, which it couldn't do as his hat flew off quite a while before, and the winds would have blown it far from the cliff's edge.