At the very beginning of the first scene in the White House, look closely at the President, his secretary and the 2 other men: they're all standing perfectly abreast of each other at the end of a hallway, as if they were just standing there waiting for the director to yell "Action!". If they really had just come around the corner (which is the only place they could have come from, since the hallway dead ends on a window), they should be jumbled more closely together, being able to walk abreast of each other only after 3 or 4 steps.
At the beginning of the meeting in the War Room at the Pentagon, while everyone is milling around and providing "small talk", general Stark mistakenly calls general Black "Whitey", when his actual nickname is "Blackie".
When the group at the Pentagon is debating whether or not to send the fighters to shoot down the bombers, Dan O'Herlihy almost jumps in with a line before he's supposed to speak, but he visibly catches himself and waits for the man next to him to deliver his line before speaking.
The UFO is reported initially "near Hudson's Bay" and its course is given (both in dialogue and on text superimposed on the map display) as 196, heading for Detroit. But according to the map display itself, it initially appears near the Labrador coast, more than 500 miles from Hudson Bay; and while it is heading for Detroit, its course from there is to the southwest, about 225.
The bombers in the film are portrayed using a Convair B-58 Hustler (see trivia entry), except during the final attack on Moscow, when the plane briefly appears as a North American F-86 Sabre Jet instead. Also some on-board footage of an X-15 experimental plane launch is used to depict the contrail of the missiles fired at the last attempt to stop the bomber (also, previous external shots of the bombers did not show much of a contrail, whereas the rocket engine exhaust of the X-15 is clearly visible).
After professor Groeteschele concludes his remarks on the casualties that NYC will suffer and the need for excavating large corporations' records (which are vital to the economy), he crosses in front of the "tactical" screen. The screen displays the sixth "defensive" bomber (which carried no bombs) and its spread of "decoy" transmissions. At this point in the film, the Russian fighters have already focused on and destroyed this bomber (to general Bogan's dismay) and the final bomber (which has no decoy capability) has dropped below radar, never to be seen on the tactical screen again. It is likely that this scene was moved during editing.
The sequence that shows the B-58 bombers taking off from Elmendorf AFB in Anchorage, Alaska, is incorrect. There were never at any time any SAC bases in Alaska, that's all Alaskan Air Command in that area. Moreover, the B-58 "Hustler" has a combat radius of only 1,740 miles, not close to the 4,000 miles each way from Alaska to Moscow. The bomber in real life, unless refueled in air by an Air Force KC-135 tanker somewhere over enemy territory (one thing the Air Force did not do), would run out of fuel not even half way to Moscow. Only the B-36 (with an un-refueled range of 8,000 miles) or the B-52 (with aerial refueling before entering enemy airspace) could possibly have flown the mission depicted in the film.
The interpreter is repeatedly described as a "translator". These are quite distinct professions (even though some people pursue both). Translators work on written texts, interpreters help people speaking different languages to communicate verbally.
The interior shots of the bombers, Convair B-58 Hustlers (see Trivia), actually were shot inside of a commercial airline simulator then under repair at a a New York airport. The three crew members sit within feet of each other, in an open cockpit layout. In an actual B-58, the world's first supersonic bomber (and capable of twice the speed of sound), the three-man crew of pilot, bombardier/navigator, and defense systems specialist were separated by banks of equipment, and had no physical contact with one another. To make survivable ejection possible on such a high-speed aircraft, each compartment was specifically designed as wholly contained clam-shell "pod" that would disengage intact if the need arose. As a result, the crew had to rely on an internal telecommunications system to talk, or a string-and-pulley system that ran along the cabin wall to exchange notes if those systems failed. It's speculated that this early "jettison pod" design was incorporated as a presidential safeguard on modern 747 versions of Air Force One, as implied in the 1997 Harrison Ford movie thriller "Air Force One", and that it also inspired the crew containment compartment of the space shuttle.
The star on each shoulder strap of brigadier general Black's jacket is attached near the outer edge of the strap. Air Force uniform regulations dictate that the single star of a brigadier general should be positioned in the center of the strap.
In chapter 9, at 40:00 into the film, professor Groeteschele (Walter Matthau) is speaking in the Pentagon. There appears to be a man dressed in black sitting under the big board, but it is the professor's shadow.
The President makes a reference to an air base named Andrews Field. In fact, Andrews Field (near Washington, DC) was renamed Andrews Air Force Base in 1947, many years before the events in the movie occurred.
When the fighters count down to light off their afterburners to try to catch up with the bombers, the shot where they light off their afterburners is actually a shot of the fighters launching missiles as you can see the missiles streak ahead of the fighters.
The Air Force master sergeant supervisor and a few other airmen in the command post are wearing their "U.S." collar insignia incorrectly. In addition, many of the airmen also wear Strategic Air Command shoulder patches on their tan Type 505 or 1505 shirts. In actuality, shoulder patches were not worn with that type of uniform.
The goof item below may give away important plot points.
After the first conversation with the Russian premier, both the president of the USA and the interpreter pour and drink water. Much later, when the last effort fails, the president offers the interpreter some water from an untouched service.