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).
Audio problem: During the chase at the end of the movie, the police cars' tires squeal in the dirt.
Continuity mistake: On the airplane, Rock Hudson takes a pillow from the stewardess, and puts it behind his head. They cut away to the stewardess, and back to Hudson, and the pillow is gone. They cut to the stewardess again, then back to Hudson, and the pillow reappears.
Revealing mistake: In this scene we see Frank Sinatra in a diving suit and helmet being lowered into the sea to look for a sunken submarine. As he is being lowered you can see the sun's reflection on the walls of a tank, meaning it was filmed in a large tank of water.