Continuity mistake: The tour group comes in and they trick the black lady into pressing the red "launch" button that displays the welcome on the screen. Just before she presses it she has a 35 mm camera hanging around her neck, but in the next shot of the group standing in front of the panel her camera is gone.
Continuity mistake: Towards the end when the NORAD door is closing, they keep showing it opened further from shot to shot in order to delay how quickly the door actually is closing vs. how slow it really would have had to move in order to allow David and Dr. Falken to actually have still made it in.
Continuity mistake: In the very beginning, when the Air Force brass land at SAC in the helicopter, they get into a 2 door Jeep Cherokee. After they enter the tunnel and stop to get out, they exit a 2 door Jeep Cherokee Chief - the wide wheel well model, not the standard one they first got into.
Continuity mistake: When Matthew Broderick is first at his computer the screen is large and has an external modem on top. Later there is a close up shot of the keyboard and the monitor. At this point the monitor is completely different and much smaller. Then shortly afterwards you can see the larger monitor again with the modem on top (behind Ally Sheedy) while it war dials numbers. Shortly after that, the monitor changes again. You can also see the changes with the color and the "side handles" used for carrying. I counted three different monitors being used.
Continuity mistake: In the scene towards the beginning of the movie when McKittrick, the General and others are discussing the problem that numerous missile commanders failed to launch their missiles because they believed the order to launch was not a test. As the camera shot goes back and forth on the actors in the room as they are talking, you'll notice McKittrick's hair goes back and forth from neat and combed to messy and disheveled.