When Juliana lifts the mask of the guy she shot in the bedroom, there is blood running down his chin. When they show it from a different angle a second later, his mask has changed position and the running blood is gone.
When the resistance attack the German soldiers, the bullets hit the ground all around the feet of the soldiers. The resistance then appears from below the road. From that position they might have just been able to see the soldiers' heads but could not have possibly shot around their feet.
Even though in English the word 'fuhrer' can correctly be spelled as such (i.e. without capitalization or an umlaut), the front page of the "Greater Reich Herald" newspaper that the doorman hands to John Smith also contains several misspellings ("furher", "Oberst-gruppenfuthrer").
The song "Tomorrow Belongs to Me" was not written until 1966.
When speaking to Juliana (Alexa Davalos) about the original propaganda films, Hawthorne (Stephen Root) says, "I was working in Canon City." The town referenced is likely Cañon City, Colorado, where locals pronounce it "Can-yon" city.