In the party scene at the apartment Tom keeps for Myrtle, there is a gramophone playing during a shot of Myrtle's sister Catherine talking to Nick. The gramophone is playing at 33 RPM. The 33 RPM long-playing (LP) record wasn't introduced until 1948. The gramophone should have been playing at 78 RPM.
When Nick meets Gatsby in his private office, the cord to the lamp on the desk changes location between shots. In shots facing Gatsby, the cord goes from the lamp to the end of the desk to his right. In shots facing Nick, the cord goes from the lamp to the side of the desk facing Nick.