When Mac is introduced to the other patients for the first time, the chess master folds the chess board such that the squares are not visible. When Dr. Suvarin takes the board from him, the squares are visible.
One can tell it is stunt doubles instead of the actors during the boat chase scene.
It's not only unlikely that high end speed boats would be commonly available in the Soviet Union, but that the ones available would have Ford logos prominently displayed on them.
When the mans hand is glued to the bucket and mop handle, he runs over to the alarm button and the alarm sounds when he clearly doesn't touch the button.
During the boat chase scene, the bottom part of either an aerial platform or helicopter is visible; it's white and is clearly off the ground.
MacGyver tells Pete Thornton he has traveled 10000 miles to get to Moscow. The distance from Los Angeles to Moscow is approximately 6000 miles.
Mac uses a wire from a light bulb to pick a lock. Using one wire wouldn't pick a lock as a person needs two prongs to force the tumbler into the correct position. This point is driven home later when Pete Thornton is later picking a lock and he uses the two prong method.