Factual error: The passenger door at the front of a 747 opens outwards and towards the front of the plane (as seen when President Marshall boards the plane). When the first family is being evacuated at the end of the movie, the door is nowhere to be seen.
Suggested correction: That's because Gibbs had blown the emergency lock on the door and it went flying. Now, whether they'd actually do that and risk the door flying through the engine is another story, but I doubt the engineers had a mid-air rescue in mind when they designed it.
Continuity mistake: When Agent Gibbs shoots the 3 guards on the airplane, the last one who is shot keeps both the arms lowered. Then, in a next shot he is with the left arm lifted up. (00:20:35)
Other mistake: When Gibbs goes into the Secret Service Room, he passes by an agent who he asks if he is sleeping on the job. He kills the other 3 agents, opens the weapons hold and walks past where the other agent was sitting, and he's now gone, and is still gone when the terrorists storm the cabin. Where did he go? Considering earlier during the tour, Melanie Mitchell warns them not to pass that point without an escort, or they'll be sorry.