When the Barber first returns to his barber shop, he hangs his hat and coat on a coat-rack that has a hand-broom hanging on it. After his fight with the Storm-Troopers, he re-enters his shop but his coat and hat are no longer on the rack, and the broom has changed location on the rack.
When Hannah has tomatoes thrown at her by the soldiers, she has 3 prominent streaks of dirt on her right cheek as she cowers down to protect herself. When she gets up after the attack, the dirt is missing from her cheek.
When the Barber flies upside down in the aeroplane, he sits at the left. He is the closest to us. Schultz asks him for the time. Near his elbow the G from the text "G 3824-R" is closest to his elbow. Later on when he gets his watch to answer the question, the plane seems to fly normally, but then the Barber sits at the right. Then suddenly it is the R which is closest to his elbow.
When Hynkel is tearing the medals off of Herring after learning that Napaloni has beaten him to Osterlich, a distant shot shows a remaining striped ribbon on the right side of Herring's jacket. In the subsequent close shot, that ribbon is gone.
The goof item below may give away important plot points.
Making his speech during the invasion of Osterlich after replacing Hynkel, the Jewish Barber quotes the Gospel of St. Luke (chapter 17, verse 21) fluently from memory, an odd accomplishment indeed for someone with the many of challenges of his religious/ethnic background, mental disability, and general ineptitude.