(at around 27 mins) 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.
(at around 1h 27 mins) 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.
(at around 12 mins) 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.
(at around 12 mins) Some of the time that the Barber and Schultz are upside down, they are backwards. They actually switched seats and the Barber is now on the right, instead of the left where he properly belongs.
(at around 54 mins) Hynkel kicks the globe balloon with the heel over his shoulder, thus the balloon is on a curved trajectory towards the window. In the following shot, the balloon moves just straight up- and downwards.
(at around 29 mins) 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.
The goof item below may give away important plot points.
(at around 2h) 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, Although not impossible, it is extremely unlikely that a working-class Jew living in a Jewish ghetto in the early 20th century would have read the Christian New Testament from which the quote is taken, let alone be able to recite it word for word..