Revealing mistake: When Griff and his gang are flying through the air and about to crash into the courthouse after the hover board chase, look at the girl on the left. Her legs hit one of the building's pillars, and her body rotates sideways, but she continues moving forward - obviously suspended by wires.
Factual error: Marty is pursuing Biff at the high school as the other Marty pulls up with Lorraine in Doc's car. In the parking lot, the car closest to Biff is a brown 1957 Chevrolet. The front grill and styling are distinct and unique to the 1957 model year. In November 1955, it would be plausible to see a 56 Chevy in the lot, but a '57 is a bit early.
Revealing mistake: When Marty is burning the almanac, during the lightening strike, the shadow of the post holding the hover-board off the ground is visible. - watch the hoverboard when the scene is illuminated by lightning strikes. The post holding up the hoverboard is also clearly visible when they pan around Marty as he is holding the burned rope from the Delorean.
Continuity mistake: When Biff, in the alternative 1985, presents the record book to Marty, he puts it in a safe behind a portrait, and then he closes it. Biff continues speaking and you can see that the portrait is slowly opening again. Cut to Marty, then back to Biff, and you can see that now the portrait is completely closed.
Continuity mistake: In the Cafe 80's sequence we see little Elijah Wood and his friend investigating the Wild Gunman game. One of them asks "How do you play this Thing?" while Marty is several feet away, but Marty answers immediately while standing right next to it. So he has managed to walk about 15 feet or so in a fraction of a second.
Continuity mistake: When the DeLorean first lands in the alley in 2015, there is a shot of the back of the car as it lands. The large vents that extend out of Doc's modifications to the car are the wrong size. In most shots, they extend over the car's tail lights, however in this shot they stop above the lights. According to commentary this is a fiberglass scale model of the DeLorean used to film flying scenes, since it is lighter than the actual vehicle, so apparently the model was not quite identical to the real DeLorean.