When Ernst Hanfstaengl listens to Adolf Hitler speak the first time you can briefly see a Swastika flag behind Hitler in one shot. It is not until later that Hitler comes up with the swastika symbol and introduces it at Hanfstaengl's dinner party.
The flag with the swastika in a white circle on a field of red was not in use in the early days. The emblem combining the swastika with the German eagle seen on a lectern was not used until the Nazis were in power.
During the segment set in "July 1918" the Lieutenant informs Adolf Hitler that the unit is to be redeployed to the Eastern Front. Russia had already signed a peace treaty with Germany on 3/3/1918 so by this point in the war there was no longer any "Eastern Front".
During the Beer Hall Putsch, Ernst Hanfstaengl is walking next to Adolf Hitler on the way to the War Ministry. In the next shot he is running away from the mob. How did he get to the back of the mob at the pace he was walking?
When Adolf Hitler is speaking to Fritz and putting forward the suggestion to write for his party he is eating a small cake. When he first picks it up he takes one small bite, when the cake is seen again there is very little left.
When the maid comes down the stairs reading from the newspaper, Helene Hanfstaengl approaches her and starts a discussion about the article. When Helene leaves the room, there is a tape mark on the floor where she stood.
Adolf Hitler would never have spit like he did in the movie during speeches at the Muenchen Bierstubl. Despite his atrocities, Hitler was known to have excellent manners and would never have talked and spit up food, as he does during the interview with Fritz Gelich.