When Otto rides his motorcycle back to East Berlin, there is a balloon on the tailpipe with "Russki Go Home" printed on it. As he rides through the streets of West Berlin, the lettering is on the side of the balloon. By the time he crosses into East Berlin, the lettering has shifted and is now on the rear.
Schlemmer calls his former superior officer in the SS "Herr Oberleutnant". The SS had no rank of Oberleutnant nor did its members call each other Herr. Oberleutnant was a German Army rank. The corresponding SS rank was Obersturmführer.
In the Grand Hotel Potemkin sequence showing the crossed Soviet and East German flags, the East German flag is upside down (black stripe should be on the top); maybe this was a deliberate error on the part of the filmmakers.
The sign that would have been displayed as you left West Berlin into the Eastern side via the Brandenburg Gate, would have read 'You Are Now Leaving The British Sector' as that area had been in The British Zone of occupation since 1945. Hence, any allied border guards or MPs that the characters would have come into contact with throughout the movie, would also be British and not American.
In the Grand Hotel Potemkin scene, Peripetchikoff says "We cannot interfere with internal affairs of sovereign Republic of East Germany." The correct formal name of the country was the German Democratic Republic.
In the scene with the three MPs, when one opens the door and says he wants to be taken away, the other soldier call him "Sarge" at least twice. However, the insignia on his sleeve shows he's a Specialist 5th Class, equal in pay grade but not addressed as a sergeant. The two soldiers with stripes (Corporals) would actually out rank the specialist who was not considered a Noncomissioned Officer.
In the shots from the west side of the Brandenburg Gate, a sign is shown reading: "ACHTUNG! YOU ARE NOW LEAVING WEST BERLIN." No such sign, in mixed German and English, was ever posted in Berlin. The signs advising that one was leaving or entering one of the four sectors of the divided city were all written in four languages: German, English, French and Russian. Apart from its factual inaccuracy, it would make no sense to post a sign written partially in one language and partially in another.