Plot hole: When Old Biff goes back to 1955 to give himself the almanac, he comes back in the Delorean to the version of 2015 that he left, not the other, skewed version in which he is rich. Everything in 1955 should have changed around the Doc and Marty, as the Doc tells Marty everything will change around Jennifer and Einstein later on in 1985, when Marty and the Doc go back to restore normalcy. George is alive in this future, so we know the skewed version hasn't taken hold.
Continuity mistake: When Marvin, the band member, goes to call Chuck Berry about Marty's "new sound", the telephone he uses has a black neoprene coiled handset cord. In the first BTTF, the phone had a straight, uncoiled brown cloth handset cord (seen when he says, "Well then listen to this." and holds the handset out toward the stage). This isn't because the cord was stretched out, as neoprene cords can't stretch so much that they look uncoiled.
Continuity mistake: At the start, the DeLorean hits the left dustbin and comes to a halt. At the other side of the street, there is only one car parking near to a house. From another angle there's a different red car parking on the other side of the street, and the tree that stood between both houses has gone. (00:01:15)
Continuity mistake: When the rain stops after Marty and Doc first get out of the DeLorean in 2015, Doc looks up at the rapidly clearing sky. Directly above their heads the sun is shining brightly. However, if you look at the tops of the buildings you can see sunlight hitting them from the sides. I suppose we will have another sun by 2015? (00:06:42)
Revealing mistake: During the tunnel chase at the end of the movie, in the shot shown from the ceiling of the tunnel, you can see in the shadow of the hover-board, skate-board wheels.
Revealing mistake: When the hovering object walking the dog in 2015 flies by, a moment before it disappears, you can see the wire used to support it.
Continuity mistake: Right at the start of the movie where Doc Brown pulls up on the drive of Marty's house, we see a panning shot as Marty and Jennifer walk to the DeLorean - there are no vehicles parked on that side of the street. A moment later as Marty waves his hand in front of the Doc's glasses, there's a green VW bus parked almost outside the house. Then as the Doc's looking through the rubbish bin, the VW's moved further down the street. Then as the DeLorean backs off the drive, there's a yellow VW bus there instead. (00:01:20 - 00:02:05)
Revealing mistake: Throughout this movie and Part III, whenever there is a special effect, like two Biffs in one scene, the scenes with the flying cars, and basically whenever an actor is playing three or four roles at the same time, if you look very carefully, you can see the picture flash ever so slightly repeatedly. It looks almost as if the picture is throbbing a bit. Might be more visible if viewed on the DVD.
Other mistake: A book that thin would not be able to hold even one year of statistics of all sports at the detail level purported in the film. When we see the pages (for example in the tunnel, when the book is in the wind screen of the car) we can see that the font is quite large, and there are even text sections - it is not all tables and numbers. And it can't just be regional results, since we hear about a college football match, which is listed in the book.