Revealing mistake: When Marty pours the wake-up juice through the funnel into Doc's mouth, in the close up the funnel is straight, but we can't see Doc. From a different angle where we do see the funnel stuck in his mouth, check the right side of the screen: The funnel is skewed (and seemingly it must have a tap) and the liquid is falling out in large amounts to avoid the actor from drinking it.
Continuity mistake: When Buford duels Marty, the people behind him differ between shots. In the shot outside, there's a tall man with a blue shirt, a young guy and a man with a white shirt. From the angle shot inside the bar, the number of people behind the group has doubled, the man in white is gone and a woman has magically appeared standing next to the man with the blue shirt.
Continuity mistake: After the modern train destroys the DeLorean, Marty lays on the grass, face down. Then, in a close-up, he turns around to watch the train. When the angle changes he is still facing down and crawling on the grass. In the following shot he has moved 5 meters to the right and is sitting by the bushes close to the tracks.
Continuity mistake: After the train crashes against the "Tracks end" sign, the editing is all wrong: Marty is inside the car covering his head with the hover board. Half a second later he has magically moved outside the car, dangling and holding the hover board. Then we see Clara upside down, and back to Marty back inside, removing the hover board from his head after the crash. then he dangles outside with the hover board. So the first shot of him dangling shouldn't have been there.
Continuity mistake: When Buford chases Marty, the position of his 3 goons is an absolute mess and changes from shot to shot with no continuity whatsoever: Sometimes they're behind Buford; or two riders on the right and one on the left; or last but not least: all three riding past Marty until a shot later one of them suddenly appears way behind the group.