Plot hole: Accessing the secure room through the duct work would be virtually impossible (pardon the pun). First is this an air supply duct, return air supply duct or an exhaust duct? If it was an air supply or a return air supply duct there would be a squirrel cage fan moving the air. Not the propeller fan you see in the frame. If it was an exhaust air duct again no propeller fan. The size of whatever duct it is for that size of a room is questionable. The room was small yet the duct was large enough for a man to crawl through. The last reason why this would be impossible is where are all the screws that ducts are assembled with? They would definitely stick through the ducts and catch clothing, scratch hands, heads etc.
Plot hole: There is no reason whatsoever for Jim to have brought the bible from the hotel with him. Max says that Job never quotes scripture in his messages, so he has no reason to bring the bible. The only biblical reference is job 3:14, and that is simply a phrase to remember: having the bible with you will not help you remember it in any way.
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)
Factual error: Luther asks for a laptop with an "AI RISC chip". This is nonsense - RISC stands for Reduced Instruction Set Computer and is a description of the number of registers used by the chip - nothing to do with AI, which is a broad range of analytical and non linear heuristic techniques to enable a computer to learn and make non-deterministic (ie. non programmed) decisions.
Factual error: In the scene in the aquarium restaurant where Kitridge reveals to Ethan that the operation was a mole hunt Ethan uses the chewing gum explosive to smash the glass and make his escape. However, the explosion also causes a man within the aquarium restaurant to go flying through the air and through the window. It is physically impossible for a chemically composed explosive of that size, that is not enclosed, to create enough force to move him in that way, add to the fact that he is seemingly the only person or object (besides the aquarium glass) affected by it, all these combine in making this scene physically impossible.
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.