The computer map of Central and Eastern Europe, as shown on the great display (ca. 7:55) has at least three errors. First, the central part of the map (including Poland, Czechoslovakia and Austria) is somewhat squished and out-of-proportions (most notably, it caused Warsaw to move significantly south from its actual position, and both Poland and Austria's borders are visibly different from their actual shapes). Second, and probably the most glaring error, is that according to that map, distance from Warsaw to Berlin and from Warsaw to Moscow is roughly the same; this locates Moscow somewhere in midwestern Belarus (!), as the distance from Warsaw to Moscow is at least 2.5 times bigger than from Warsaw to Berlin. A minor error is that Russian (now Ukrainian) city of Kharkov is misspelled as "Kharov".
The view of the aircraft carrier is reversed, as the angled deck is on the wrong side of the ship. Also, this makes the ship's hull number 18, which was the USS Wasp, not the USS Midway (CVB-41). If the image was not reversed, the hull number would be 81, which has not been built as of the year 2007.
Once the Secretary of the Interior boards the E-4 and assumes the duties of president, the plane should have been referred to thereafter as Air Force One, but at no point in the film is it ever referred to as anything but the E-4. (It is correct for Alice and his crew, however, to revert to calling it the E-4 once they learn the original president is still alive.)
In the opening scene, the various military alert levels are referred to as "ROUND HOUSE," "COCKED PISTOL," etc. These designations are only used during exercises, to ensure that there is no confusion as to whether an actual change in alert status is being ordered. However, in an actual crisis, such as the one depicted, the statuses would be referred to by their DEFCON (Defense Condition) numbers, from DEFCON 5 (for peacetime readiness) up to DEFCON 1 (for maximum readiness, i.e., war).
In the cast credits, the character that reads "Lt Hooker" should read "Sgt Hooker"? After Maj O'Toole's death, Maj Cassidy tells Sgt Hooker that he has just received the first battlefield commission of this war, which means he is now a Second Lieutenant.
During a scene aboard Air Force One, the clocks on the wall
(supposed to depict the times of different major cities around the world) all show vastly different times. Time zones by and large only differ on the hour, the few half-hour zones including Delhi and Newfoundland, so the minute hands on the clocks should at least mostly all have been in the same position, with only the hour hands being on different positions.