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.
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.
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.
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 train is running on the right-hand track, but in the UK and France trains run on the left-hand side.
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.
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.