Factual error: When they launch the F-16's to intercept the Soviet "Backfire" Bomber, the picture shown is of two F-15's. In addition, no F-16's were stationed in Alaska to handle air defense duties, this was handled exclusively by F-15A/C's.
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: Near the end of the movie, the panel has JPE 1704 TKS showing, but when Joshua finds the launch code, CPE 1704 TKS is blinking on the screen.
Factual error: During the part where they're in NORAD and watching all of the Soviet "nukes" hit various bases in the US, a base in northern Maine is listed as, "Loring AFB 43 Bomb Wing." This was a real air force base, but it was the 42nd Bomb Wing that flew B-52s out of Loring. The 43rd Bomb Wing did exist, too, but it was based at Andersen Air Force Base, Guam, in the North Pacific.
Factual error: At the end of the movie WOPR tries to crack the launch-code using brute force. So far so good. When WOPR finds out one digit of the 10 digit code, the first digit locks and the search goes on with the remaining 9 digits. Then he finds the second one, it locks too and so on. Problem is, brute force doesn't work that way. It would be too easy (26 letters and 10 numbers = only 36 possibilities for one digit). Brute-Force works only "all or nothing", you can't sneak your way to the whole code one by one.
Revealing mistake: Where the officer is pushing the buttons on the wall panel to launch the missiles, on the both occasions you see it, the panel buckles highlighting the flimsy set.