When Jim Phelps, giving his name as "Roger Benedict," sends Col. Sabattini (John Vernon) a diamond from the nation's crown jewels, Sabattini tests whether the diamond is real by using it to carve a scratch on a mirror inside a wooden box. When the scratch line is shown a moment later, however, it is a different length, in a different location on the mirror, and cut at a different angle.
It cannot be proven that a diamond is real by whether or not it scratches glass - any number of a large variety of materials can do that with relatively small amounts of effort, including ordinary rocks and plastic toys.
When Barney, being at this stage blind, drills the hole into the crypt and inserts the oxygen tube, he turns on the oxygen. However, the tube with the oxygen is shown as being twisted at right angles to the valve, which would cut off any oxygen. If there was any gas flowing, it would have forced the tube out to a reasonable angle. So there was no oxygen gas flowing.
When Francesca is put into the mausoleum, she apparently runs out of air in around 3 minutes. An average coffin can hold 5 hours of air.
Willy (Peter Lupus) has been transporting the archbishop (Marcel Hilaire) to the palace by donkey cart, when they meet up with a soldier on a motorcycle with a sidecar. The motorcycle takes off at high speed to get the archbishop to the palace to perform the wedding. Yet moments later, Willy is seen inside the truck parked outside the palace where Jim (Peter Graves) is readying his equipment, while the archbishop, despite traveling directly to the palace at high speed, somehow does not arrive at the palace for quite some time.
When Paris double-masks, he has blue eyes as the prince. Logically, the disguise would use colored contacts, but since Paris is about to cover the "prince" with a mask of his brown-eyed self, there would be no reason to don blue lenses at this point.
When General Sabattini, sitting on the desk, hangs up the telephone he misses the cradle.