Visible crew/equipment: When Paul Newman takes Julie Andrews up a small hill to explain that he is not going to work for East Germany you can see a shot from a distance. It is clear that this scene was done in a studio, as you can see several lights above the blue background (which is supposed to be the blue sky).
Continuity mistake: When the German agents dig up Gromek's buried motorcycle and are pulling it out of the hole in the ground, it is completely clean and in perfect condition with no dirt, scratches, or dents on it.
Continuity mistake: When Julie Andrews and Paul Newman are in bed at the start of the movie they have a white phone next to the bed. Shortly afterwards when Julie answers the phone, it is an old-fashioned device.