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.
Plot hole: After Biff steals the DeLorean and travels back to 1955 with the Sports Almanac the movie states that he creates an alternate timeline. Doc Brown does a good job of illustrating this on a chalkboard. So if, as Doc Brown states, Biff created an alternate timeline, then when he travels back to 2015 it would be the alternate 2015. Even if the original timeline stays intact with Doc and Marty in it, they'd be trapped there because Biff took the Delorean and returned it to the new timeline, not the one they're in.
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.