It's raining when Nick drops off Daisy and Tom after being in New York. Gatsby, who has been standing in the driveway for a while, is completely dry.
When Myrtle is banging on the window, her fingernails have bright red nail polish. After she breaks the window, and is bleeding, her nails are bare.
When Tom smacks Myrtle, her shirt is off her shoulder on one side. In the next shot, when blood is coming from her nose, her shirt is back up where it belongs.
During the second party scene, as the Charleston begins but before the swimmers run in, Nick is clearly visible on the left side of the screen, dancing in his outfit from the first party.
When Jay shows Daisy his closet, the shirt in Daisy's hands changes from pink in a close-up to yellow in a long shot and back again.
When Tom Buchanan and Jay Gatsby are about to drive into the city, Tom asks Jay if he can drive a stick. All cars had manual transmissions in 1922; GM introduced the Hydramatic, the first automatic transmission in a production car, in 1939. In the source novel, Jay asks Tom if he can "drive a stick shift."
The puppy that Tom buys Myrtle ages at least 6 months by the time it gets to the apartment.
When George Wilson removes his gun from the paper bag, the gun is clearly not loaded.
When Nick and Jordan walk out from the pedestrian underpass into Central Park, an air conditioner is visible in a fourth floor window of the office building in the background. Window air conditioning units were introduced in the 1950s.
In the party at the apartment Tom keeps for Myrtle, a gramophone plays at 33 RPM during a shot of Myrtle's sister Catherine talking to Nick. The gramophone should have been playing at 78 RPM. The 33 RPM long-playing (LP) record was introduced in 1948.
One shot includes a cardinal and a house finch at Nick's bird-feeder. House finches were introduced to Long Island in the 1940s, before that, they were found only in the western US.
The story is set in 1922, but Gatsby drives a 1928 Rolls Royce Phantom I.
Newport Bridge, which appears in the background of some wide shots,
was built in the late 1960s.
The billboard in the Valley of Ashes misspells the word oculist "occulist".