Continuity mistake: When Tom and Nick get off the train and go to Wilson's shop, in a close up of Tom's face you see him look up toward the window (looking for Myrtle). After that it cuts to a wide shot and you can see Tom in the background repeating what he did in the close up.
Other mistake: At the very end Nick writes "The Great" in ink above the typed title "Gatsby" on the cover of his manuscript. The E in "The" is a completely different style to the E in "Great."
Suggested correction: It is the therapist who writes The Great. As for why the Es look different, not everyone one is able to write their letters the same every single time.
Continuity mistake: When Tom discovers that he is going to be abandoned by Daisy, he extinguishes his cigar. The grip swaps from holding the tip to being on the middle part of the cigar, between shots.
Continuity mistake: When Nick meets Jordan, she is sitting on the sofa holding a magazine with both hands, yet less than a second later she manages to be lying horizontally.
Continuity mistake: When Gatsby finishes throwing the shirts at Daisy, the type and amount lying on the floor changes between shots.
Continuity mistake: While Gatsby says the "you can repeat the past" speech, Nick is standing against a column. Then the shot changes to a wide angle where he starts to walk away. Then, suddenly, the previous angle is used and Nick is back next to the column for the rest of the scene.
Continuity mistake: While driving the yellow car, Gatsby flashes his business card at the cop, whose head swaps between looking sideways or straight ahead depending on the angle, no continuity whatsoever.
Audio problem: After the barber shop scene, Jordan tells Nick about the tea date. While she talks, there's a side angle where her lips are totally out of sync with the audio.
Continuity mistake: Early in the film Tom brings Nick to meet with his mistress Myrtle - and set Nick up for some drinking and debauchery. While waiting for Tom to come out of the bedroom Nick is sitting in a small parlor with Myrtle's dog. In the first wide shot we see the dog licking an empty plate. Cut to an expression shot of Nick staring awkwardly, then a close-up of the dog - but now a small pile of dog food has appeared on the right side of the plate. Another reaction shot of Nick, then back to the dog - but the food has moved to the left side of his plate.