An axe disappears off a wall between shots during the craps game. Chaplin originally shot a gag using the axe (photos of this sequence exist), but cut it from the final film, which created a continuity error.
In the restaurant scene, after the customer is beaten up and thrown out for being 10 cents short, his hat is seen on the floor next to the cashier's desk. When the waiters come back in, the hat is gone.
At 3:57 Charlie lets the girl take his seat at the table. The cook gives her a dish of soup. Right after Charlie opens the door, the dish is gone.
When the ship arrives in New York harbor, it is moving in a southward direction as indicated by the passing view of the Statue of Liberty, but a ship arriving in New York would be heading north, not south.
At around 21 minutes, after the huge waiter, Eric Campbell, bends the lead coin in his teeth, Charlie asks for more coffee. Just before Campbell exits, while taking Charlie's cup he says something that makes Edna Purviance break character with a smile. She covers her mouth and turns from the camera to hide her reaction.