Continuity mistake: When Agent Gibbs goes into the closed room and hands out the advance team reports from the Miami field agent, when he executes the three agents, if you watch, just after he kills the third agent, the agent's entire arm falls down behind the arm rest below the seat. However, in the following shots, when Gibbs re-holsters and turns to enter the armory, the agent's arm and hand are now risen upward, leaning against the window with his elbow propped up on the arm rest. (00:20:45)
Continuity mistake: In the scene where the President is being escorted to the escape pod, there is an extra shot where the President is hiding on his own around the corner from a terrorist, who is sneaking around in the background looking for him. In both the preceding and following shots, the President is being physically escorted by two Secret Service men who are pulling him along and forcefully holding his head down. This shot is wrongly inserted into this portion of the scene and needs to be later in the film, when the President is trying to escape. (00:23:40)
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)
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.
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.
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...