The person playing the piano is not playing the correct notes when the Russian's are doing their gymnastics near the beginning of the movie. Maybe it was recorded with one music, but the new music that was dubbed on the soundtrack is not the same that the person is playing.
During the car chase, when Bruland (Donald Sutherland) is supposedly driving from the back seat, the car does a 180 in reverse and a brief shot from outside shows the actual driver is driving and not Bruland, and you can see his arms spinning the steering wheel.
When Griff and Bruland and the Russian gymnast are escaping by jumping over a railing, the stair landing is obviously cushioned.