Factual error: The vents that Hunt and his sidekick crawl down at CIA Headquarters are standard galvanized steel box vents; they are very common in the building trade. Try walking or crawling down one - you'll make a noise like the sky is falling down. People will be able to hear you for miles. Every person in that building would know somebody crawling about in the vent system. (This error applies to dozens of films, not only this one).
Factual error: CIA headquarters in Langley has its own firefighting unit and never, ever calls for outside help. In fact it is official CIA policy that they would rather have a building and its contents burn to the ground than allow unauthorised access by firefighters they know nothing about.
Factual error: The train is a French TGV, these do not operate in the UK (not to date anyway) The Channel Tunnel passenger train is the Eurostar.
Factual error: When the helicopter blows up, the train stops dead in its tracks almost instantly. A high-speed train like the TGV needs more then two miles to come to a complete stop from its top speed, or at least from the speed it's traveling at. And a Little Bird attached to it wouldn't make the slightest bit of difference.
Factual error: Max@Job 3:14 is an illegal e-mail address.
Plot hole: In the air vent, we see a rat. Wouldn't rats in the vents set off the alarms on a regular basis, because the lasers were on the upper side of the vent?
Continuity mistake: In the scene where Ethan is in the NOC list room Krieger starts to drop him and Ethan is hanging there for a little while. Suddenly some sweat starts to drip down his glasses and he catches the drop with his hand. If you look at how close he is to the floor this move would be impossible.
Factual error: In the scene where the helicopter is being flown into the tunnel, sparks can be seen on the far tunnel wall, where the rotors have hit it. No helicopter can keep flying in a straight line after the rotors have collided with a solid object in this way.
Continuity mistake: Besides the fact that during the final action sequence it is impossible for two trains to occupy the same tunnel, how is it that the blades of the helicopter attached to the first train don't strike the second? Tight fit.
Continuity mistake: The train is electric, supplied by overhead cables, which do not appear to be present at any time. If it was a Eurostar then it could use 3rd rail pick-up but it wouldn't be going so fast, but that's academic; its not a Eurostar and no 3rd rail is visible either.
Factual error: When Ethan is lowered into the computer room at Langley, he is dropped and nearly lands on the pressure-sensitive floor. When an object moves through the air, it pushes the air in front of the object. As the floor sensors were set so high that a drop of water was sufficient to set an alarm off, the air displaced during Ethan's fall would have triggered them too.
Factual error: The Channel Tunnel tracks run through separate tunnels, one going to France, one going to Britain, so trains don't pass in the tunnel as they do in the film.
Continuity mistake: At the end of the film, Ethan drops his mask. His hair looks flat and wet. At the next shot it's dry and up. (01:34:20)
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.
Factual error: The train is running on the right-hand track, but in the UK and France trains run on the left-hand side.
Revealing mistake: When Ethan uses the magnetic screwdriver to undo the screws holding the grille to the air-vent in the NOC list room, the spindle that is actually turning the screw is visible after it falls out.