Continuity mistake: When Harrison Ford tries to enter the conference room when he comes upstairs from the lower level, he first struggles with the guard to the conference room door. While fumbling with the keys, the guard "wakes up" and shoots 2 shots toward Harrison Ford, where he turns around and shoots back. Those 2 shots from the guard leave bullet holes in the conference room door. When Harrison finally enters the room, the bullet holes disappear. When the bad guys return, the bullet holes are gone.
Continuity mistake: At the beginning, when AF1 is landed, and the two pilots and navigator get shot, a Russian guy takes over. AF1 is then driven around the airport. The Russian guy controlling AF1 speeds up to take off again. When he does take off he barely misses a plane that was on the ground. The next shot is from the back of AF1 and the plane that was on the ground has disappeared.
Continuity mistake: When Harrison Ford is fighting one of the terrorist guys, they go into a room with two chairs. Ford spins a chair to knock down the guy, and they keep fighting. As they go at it we see a knocked over stool with a black seat. Instantly, in the next shot, the stool is standing up, and in a perfect position for Ford to knock the guy out with. (00:44:30)
Visible crew/equipment: When Ford and a terrorist are fighting, just before Ford knocks him out with a stool, the shot changes to their tussle where the gun is knocked away into the aisle. Watch this in slow motion and you can see the hand throwing the gun into shot. (Visible only on fullscreen version)
Continuity mistake: In the scene when the Russian is shooting the 3 doors, on the second door, if you look in slow motion, you see 2 muzzle flashes before a hole is made in the door. The same on the 3rd door. Also, when he shoots the 3rd door, he puts 5 bullet holes in it, but when he opens it to check, there are 6 holes in the wall.
Factual error: In the scene where AF1 gets refueled by a KC-10 Extender tanker, it would be impossible to pull the whole refueling probe out of the tanker plane, not to mention causing a fire. AF1 would simply break the end of the probe, the tanker plane crew would close the internal fuel valves, and there would be no fire. The refueling hardware used on USAF tanker aircraft has predetermined breaking points, since it's likely that any other aircraft getting refueled suddenly loses height, gets off course or something.
Continuity mistake: When the traitor first begins to shoot the 3 people, we see a close up on one of the victims (the one who gets shot in the head). If you look behind him in slow mo, you'll notice that the furniture is already red (from the blood).
Trivia: A little aside - when the terrorists first strike, in amongst the confusion you can hear someone shout "get the football!" The "football" is the name used for the case which contains the nuclear launch codes, but this is never explained. It's not a major thing, but interesting to note that they included it as a little detail. Good to know a childhood spent reading Tom Clancy books wasn't wasted...