During the chase scene in 'The Beast' Sam has to use satellite view to drive backwards because the rear-view mirror was destroyed. At the end of the chase there is a shot of Sam's eyes reflected in the rear-view. Likely this scene was originally intended to take place earlier in the sequence.
When Will Derringer has the udder from a sheep in his face it shows four nipples (like on a cow udder). Sheep only have two nipples on their udders.
The terrorists use a cargo carrier to intercept Air Force One, but the aircraft always flies with a fighter escort over international airspace. That aircraft would have been shot down.
When Prime Minister Clarke is receiving his audio briefing during his morning run his staffer refers to him as "Mr. Prime Minister." In the UK (and other Westminster systems/Commonwealth realm nations) there is no gendered prefix given when addressing the head of government (unlike the American president who is referred to as "Mr." or "Madam"). They are simply referred to as "Prime Minister."
During Prime Minister Clarke's morning brief he is told that the Economic Minister would release a statement regarding unemployment. This position does not exist within the British Government. All economic matters fall within the remit of the Chancellor of the Exchequer, who would be referred to as such in any Government briefing.
Twice, reference is made to "enriched plutonium"; plutonium is produced in a reactor and is mostly Pu-239 which (unlike natural uranium) does not need to be "enriched". Also, it is implied that a pebble bed reactor requires plutonium as a fuel, but this is not the case.
During the train ride to Trieste, PM Clarke says a couple if times, that he was 'running for office'. This is incorrect as The office of Prime Minister is held by the head of the political party that wins a general election. Clarke would have become a member of Parliament for a constituency area in the UK before being voted as head of a Party, so not elected personally to office. However, when explaining the situation to the American President ( that he was following that path ), he might well have used the nearest-equivalent term, to help the President understand.
Derringer brags about Air Force One being a "new" aircraft when it is actually a specially configured Boeing 747 that have been produced since 1969.
However, the USAF has ordered a brand new updated 747 to serve as Air Force One in 2015. If, in the film's timeline, that's the plane being used, then it would qualify as new.
Sam brags about vetting Will but apparently didn't know his middle name.
Flipped shot: The press photographing Sam and Will boarding Air Force One are using cameras that have the shutter release on the wrong side.
When Air Force One is rolling over and the oxygen masks hang down from the ceiling, the Minister and President are rolling up the wall and across the ceiling, oxygen masks dont change direction, keep hanging down.
When the bad guys enter the safe house, all their guns have lasers and all are red. In a camera change, and for a couple of seconds, all the lasers become green and in the next scene they are red again.
Although western Belarus (the area around Hordna as shown on the Air Force One map) is a bit hilly, there are no forest-covered mountains like the ones depicted in the film. The land there is almost entirely developed for agriculture with only a handful of groves. The Appalachian or the Carpathian Mountains were likely used for those shots.
There's no way a wide-body Boeing jet can break Mach 1 speed. It is structurally unfit to bear this kind of aerodynamic force. However, perhaps President Derringer is exaggerating or has made a conscientious mistake.
Noelle Bisset pronounces tomato "toh-may-toh" in a very un-British manner at the beginning of the film despite being a British character.
In one of the first scenes, the American president's young daughter -- riding on his shoulders -- says the word "anything" with a distinctly British pronunciation (sounding like "EN-ih-thing") rather than the expected American accent (more like "EN-ee-thing") early in her lines. This phonetic detail reveals that the child actress is British, despite portraying a character who is supposed to have been raised in the United States by American parents.