Some establishing shots of Manhattan show the Chrysler Building and the Empire State Building under construction. The film takes place in 1922. Construction on the Chrysler Building began in 1928 and on the Empire State Building in 1930.
The story is set in 1922, but Gatsby's yellow car is a 1929 Duesenberg Model J, Tom Buchanan's blue coupe is a 1933 Auburn 12-165 and almost all of the cars featured in the movie are incompatible for the time period.
The "French" phone shown as they talk about the intrusive unwanted guest calling Tom wasn't invented until about the mid-30's. In 1922, you still had to hold the candlestick phone in one hand and talk into the mouthpiece while holding the earpiece up to your ear.
When Gatsby was at Daisy's Louisville home, the opening shot showed him with two ribbons on his chest but in subsequent shots the ribbons were missing. Beyond that, one of the ribbons was the World War I Victory Medal that was awarded to soldiers after the war and clearly this scene was before Gatsby shipped out to France, so there is no way he would be wearing it at this time.
When Gatsby and Tom take the train to the Valley of Ashes, a steam locomotive is shown, but the Long Island Railroad's Port Washington line which runs from the Island's north shore through Flushing (the actual location of the ash dumps) was electrified in 1918.
The story is set in 1922, but the music playing over the main party is George Gershwin's "Rhapsody In Blue". Even though it is overlaid rather than diegetic, it's inclusion is inaccurate as it was not written until 1924.
In the song "No Church in the Wild" there is a reference to the Rolls-Royce Corniche. This particular model was first produced in 1971, but was first designed in the 1930s (though production was halted because of WWII) while the movie is set in 1922.
The vinyl pool floats at the party weren't available in 1922; flexible vinyl was discovered in 1926 and it was years before it was used in commercially available items. At best, they might have had inflatable rectangular rubber-coated canvas floats; none would have been brightly printed or colored or shaped like swans.
In the first dinner party, Daisy comments that she's heard a rumor that Nick is to be married to a girl out West. Nick responds, "It's a libel!" As a writer, Nick should have known that the appropriate term is "slander." (Libel is written untruths, slander is spoken untruths.)
Gatsby gets completely drenched at 4:00 pm when Daisy arrives at Nick's house for tea. When he drops the clock the time in the clock is 4:20 pm but his suit is dry. It is impossible for a suit to dry within 20 minutes.
When Nick rides with Gatsby in his car, Gatsby passes a truck and waves his hat to him. In the next shot as they make a turn, his hat is on his head. In the next shot, it's in his hand waving to the truck again.
When Tom, Jordan and Nick are driving away from the accident scene the close-ups show Jordan sitting up straight with Nick's arm resting along the back of the seat, but the long shots show Nick's arm loosely around Jordan's shoulder's and her head resting on his shoulder.
Tom and Nick are at the Gatsby's party and standing at the rail on top of the stairs. As Tom leaves, Senator Gulick is making his way towards Nick from behind but in the next shot, the Senator is nowhere to be seen and all of the guests in the background have changed.
When Nick, Daisy, Jordan and Tom are in the parlor, the head footman enters and lifts his arm to announce dinner. His arm is lowered during Daisy's close-up, and then raised again in the overhead shot.
When Daisy is about to marry Tom, she pulls off the $350,000 pearls he bought her and they scatter all over the floor. An expensive pearl necklace like that would feature individually knotted pearls that would minimize loss of pearls if the silk were to break.
The tenor saxophonist at the Gatsby's mansion party is holding the saxophone incorrectly. The proper way to hold a saxophone is with the left hand on top and the right hand on the bottom. The player in the film is holding the saxophone with the right hand on top and the left hand on the bottom.
When Nick is finally invited to the party, he tries to find Gatsby but is told by several party-goers that no one has ever seen him and they don't know if he actually exists. Later in the movie,there are flashbacks of newspapers with front page stories of Gatsby buying up New York with several showing his picture as part of the article.