Continuity mistake: When Doc has just consumed the 'wake up juice' and runs out of the bar and dunks his head in the horse trough, Marty and the bartender come out after him. You can see that Doc is now soaked and a trail of water is on the ground behind him from a previous take of Marty dragging him back in the bar, which hasn't happened yet at this stage. Only really noticeable in the cropped version. (01:20:05)
Continuity mistake: When Doc and Marty are chasing the train on horseback, Doc gets on the train first and his horse pulls forward. Then when the camera shows Marty on his horse, another horse can be seen in front of him. When Marty jumps on the train, his horse falls back. But in the next shot you can still see the horse as it moves off camera. (01:30:55)
Factual error: In the scene where Doc and Clara are playing with a telescope at night, the moon is seen twice. The phase of the moon appears to be a waning gibbous. Though the phase of the moon on the night of September 5th, 1885 at pacific time was waning, it was actually a waning crescent, a much different looking phase of the moon. I looked this up on the United States Naval Observatory website.
Factual error: Sierra #3, the locomotive used in BTTF 3, is a 1891 Rogers, despite this movie being set in 1885. (00:40:45)
Factual error: The steam locomotive in the film is of a 4-6-0 wheel arrangement and is lettered as Central Pacific no. 131. The Central Pacific did have 4-6-0 engines in 1885, but according to the company's locomotive roster, the number 131 belonged to an engine of a 4-4-0 wheel arrangement. Also when the locomotive is uncoupled it shows a Janney (fist style coupler) - that style wasn't invented until 1890. The locomotive would have used a pin and link style connection.
Factual error: In shots of the town outside of the saloon in 1885 a modern day California flag can be seen. This flag was not used until 1911.
Factual error: Standard gauge for railroad tracks in 1885 was 4ft 8 1/2 inches. The distance between the center of the wheels on a Delorean was 5ft 3 inches. The car would not fit on to the tracks, and we can see the wheels aren't recessed or otherwise altered.
Revealing mistake: At the festival, after Marty is given the gun he encounters Seamus and Maggie. Due to the Vistaglide camera effect where Marty was later added to the scene, she is unable to look at where Marty is supposed to be. This is very noticeable when Seamus says "buffoon" and Marty turns around and starts to walk, but Maggie keeps staring at where he was. Also when the scene ends and she looks at Marty and warns him to watch for the future, she's totally missing the spot.
Continuity mistake: When the Doc enters the destination time into the time circuits when he is sending Marty to 1885 from 1955, the time/date etc. Go in as he's typing it. However, in the previous films and later in the 3rd film when traveling back to 1985, the enter button must be pressed before anything appears for the destination time.
Other mistake: When Clara is waiting for the train to stop at the train station and the train has steam coming from in front of the wheels, as it passes by the camera it jolts the camera. (01:20:20)
Continuity mistake: In 1885 Hill Valley, the hills are shown to be almost directly behind the courthouse, however in the previous films there was some distance between the two. This was likely due to the different shooting locations. (00:26:40)