Continuity mistake: Just after Gatsby asks Nick his opinion of him, there is a cut to a low angle of him driving. In the shot from behind the car, the number plate is on the right hand side of the back of the car. This is only visible for a fraction of a second, so you have to be quick when you look at this. 2 shots later, the number plate has moved to the left side of the back of the car.
Continuity mistake: During the dinner scene at Tom's, a couple of shots before Tom's phone rings and he goes to answer it, Nick's wine glass is about half full. After Daisy leaves to go to Tom and the shot cuts back to Nick, the level of wine in his glass has notably decreased, without him touching it.
Continuity mistake: When Nick is telling the doctor about his new cottage near the start, he goes to his porch with a book and stands next to one of the pillars. In a wide angle, he is standing next to a pillar with leaves around the pillar, but in a sudden close-up the leaves around the pillar have vanished.
Factual error: The movie takes place in the summer of 1922. They played Rhapsody in Blue, which wasn't written till 1924. The bandleader is obviously Cab Calloway. Cab Calloway's first recordings with the Missourians weren't until 1928. "Oh! You Have No Idea" also wasn't recorded until 1928, and it was sung by Sophie Tucker, not Josephine Baker (although they do credit Ms. Tucker in the final credits).
Factual error: Trains of various types appear in several shots but the trains shown - likely all CGI - are completely unlike any trains that were in use in New York in the 1920s or in any other era. They appear to be based on Australian prototypes and their design differs significantly from American trains.