Continuity mistake: When Marty is being chased by the Libyans, a few times it cuts away to show the instrumentation in the car. The first time the odometer reads 33064.2 and the trip counter says 88.8; the second time it's goes back a little to 33061.8 and 86.4. A few seconds later we see it as he turns the corner, and it goes down again from 33,061.8 to 32,994.4, the trip counter has gone from 86.4 to 19. In fact it's a completely different set of instrumentation - the bottom of the needle's angled in some shots and flat in others.
Continuity mistake: When Marty is writing the letter to Doc before he goes into the future, the word "disaster" is the only word in the last line. Yet when Doc shows Marty the letter when it has been all taped up, "terrible disaster" is the last line. Also, the handwritring in each letter is different (i.e. Marty's signature). They are two different letters.
Continuity mistake: When Marty is standing over his father's bed in 1955 persuading him to date Lorraine, there is a hair-dryer in Marty's belt that seems to disappear between different frames (note the hair dryer's cord leading to Marty's right hand, as he holds it behind his back). Really obvious. [This scene was originally longer, including a part where Marty removed the hair dryer from his belt. Still a mistake, but there's the explanation.]
Continuity mistake: When Doc and Marty are at the Twin Pines Mall and Doc is telling Marty about the car etc, he says "never mind that now, never mind that now..." and Marty has the camera facing the ground, yet later when Doc in 1955 is watching the video, you can see him saying that on the video.
Visible crew/equipment: When Doc Brown loses his footing on the clock tower as it begins to toll, in the next shot, when he swings about towards the clock the thin black stunt cable attached to his waist, that leads around the neck of the statue, is visible. Do not confuse it with the thick silver electric cable Doc is holding.
Continuity mistake: At the end of the movie, when Doc Brown leaves Marty at his house, he then backs up the DeLorean, turns the car around and passes in front of Marty doing 35 mph at the most, but just as he leaves the frame you can hear the swishing noises the car makes when it travels through time. No way could Doc have reached 88mph.
Continuity mistake: When Marty chases after Biff on the borrowed 'skateboard', Marty is wearing a dark grey belt and a red/blue print shirt under his red/beige jacket. However, when Marty is hanging on to the front of Biff's car as they turn a corner (and in another shot), Marty (stunt double) is wearing a light brown belt and solid tan shirt.
Factual error: When Biff is chasing Marty (on the 'skateboard') around the Town Square, you can see curb cuts at the corners. Those didn't come into being until the 70s.
Continuity mistake: In the Cafe of 1955 Marty goes into the phone booth to look up docs name in the phone book, when he finds it the name is on the left hand page yet when he tears it out of the book he tears out the right hand page. One could say he turned the page to tear it right-handed, however when he is asking the waiter if he knew where the address was, he is reading the address on the same side he tore out - check the tears on the edge.