When Jim starts the mission tape, there is almost no tape on the supply reel. During playback there is quite a bit of tape on the supply reel.
While the fail-safe device is being opened, the 'wrist' motors on the mechanical hands used to turn the combination lock are so loud the tumblers could not have been heard over the noise.
As Rollin pulls the front panel off the fail-safe device, a wire harness just falls out, indicating it was never connected.
When they are trying to open the device they are using small microphones to listen to the clicks of the combination lock. The "microphones" are clearly made of glass and are, in actuality, miniature light bulbs.
Near the end of the mission, Rollin still disguised as Shipherd, enters the back of a black car. Crew and lights can be seen in the reflection of the back window.
It was never explained how Rollin was able to get a complete facial impression of Shipherd in order to make a perfect mask for him to wear.
The plane crashed "last night". There wouldn't have been sufficient time for the agency to comprehend all the particulars and brief Jim.
The entire story is about a U.S. bomber that crashes behind the Iron Curtain and a brilliant U.S. scientist who defected to the unnamed country who is supervising efforts to take the device apart, which is supposed to yield valuable information about the entire U.S. defense system, but because its Fail-Safe device failed to self destruct, it must have been defective and would have failed to reveal any valuable information.
During the briefing in Jim Phelps' apartment, at one point it is revealed that the front door is open. This is highly unlikely due to the sensitive nature of the briefing and the need for secrecy.