Continuity mistake: In the scene where Harpo is climbing out of the brig and goes to the open window of the famous aviators, he is shown climbing in the room from the outside with the rope he's climbing on draped across his chest. When he gets into the room the rope has disappeared!
Continuity mistake: In the contract scene towards the beginning, Groucho and Chico, after the second part, start tearing away subsequent parts without reading them aloud. They tear off one because Chico doesn't like it (the third), one because Groucho won't like it (the fourth), and two because Groucho's word is not good enough for Chico (the fifth and sixth). At this point they should be at the seventh part, but when Groucho reads again he begins with "the party of the eighth part..."
Continuity mistake: When Gottlieb discovers Driftwood in his opera box and the latter escapes by climbing to another, Gottlieb leans against the wall and calls him a "Schweinhund." The shot then cuts to the orchestra, but the box (with Mrs. Claypool in it) is still visible and Gottlieb has somehow vanished.
Plot hole: When the three Marx Brothers are causing mayhem at the performance of "Il Trovatore," although we see scenes from several different acts of the opera, there seems to have been no intermission (or all the trouble would have gotten sorted out then, instead of at the end). You have to be at least a little familiar with "Il Trovatore" to notice this.
Continuity mistake: When Driftwood enters the stateroom for the first time, he dances into the room, with his cigar is in his mouth, but in the next shot, showing Driftwood still dancing into the room, his cigar is in his hand. (00:31:10)