Continuity mistake: As the firefighters are entering Langley they mention they need to shutdown the air conditioning. The guard states that the aircon is in sector 21 that has no alarm. Luther 'helps' by triggering the alarm in that sector. A while later when Kittridge is talking to his people in the ops room you can see that the alarm is active in the 3 sectors as the guard mentioned but now sector 21 is missing. (00:57:30)
Plot hole: There is absolutely no way Jim Phelps would have "died" when he got shot even if he was faking it. The first time we see his death he's pointing his gun right at the camera on his glasses which are on his head and not his chest. Later when we see the how he fakes his death he's not pointing the gun at himself anyway so he wouldn't be able to get the angle of the gunshot from his camera's point of view as we saw the first time.
Factual error: The TGV (and also Eurostar) trains are powered by an overhead power transfer, so an escape from the train with a helicopter would indeed be a Mission Impossible.
Plot hole: In the famous raid on the computer room raid, where Hunt is lowered from the ceiling, we see a close up of the screws to the vent being undone and caught in a holder so they don't fall on the floor. But the screws cannot be put back in their holes and done up once the vent is closed, and there is not enough time for Hunt to be pulled back up and the vent closed and screws replaced when the operator returns to the room, as Hunt was pulled back up with seconds to spare.
Suggested correction: They would not have to use screws to reattach it. They could use tape, glue, or magnets.
Continuity mistake: When the "firemen" (Ethan Hunt, Claire Phelps, Luther Stickell and Franz Krieger) leave Langley, Claire is in a red coat and red dress. As we see all four drive off in a fire truck, Claire is in a fireman's costume. Firstly, when did she have the time to change and secondly, why change? (01:08:00 - 01:10:00)
Revealing mistake: When we see a close-up of a knife being pulled out of a man's stomach, it's obviously a prop knife where the blade goes into the handle, as we can see the end of the blade while it's being pulled out. (01:19:50)
Continuity mistake: During the Job 3:14 scene, Ethan places the Bible parallel to the laptop and the gun. In the close-up angle it always appears that way, but in wider angles the laptop is slightly slanted. (00:34:15)