Factual error: When they launch the F-16's to intercept the Soviet "Backfire" Bomber, the picture shown is of two F-15's.
Continuity mistake: During David Lightman's escape from the infirmary, notice the hands of the guard officer - he's wearing his gloves, and the next instant they have disappeared, and reappear in the next scene.
Revealing mistake: In the scene where David starts the game with Joshua, it asks him what side he wants to be (1 for the U.S. and 2 for USSR). Look carefully at the closeup of the computer screen - his choice, 2, appears long before David actually presses the key.
Audio problem: When Jennifer gives David a ride home near the beginning of the movie, you hear David's dog barking. In the next shot, we see the dog is running down the front path and it barks a couple more times. Look closely at its mouth, it's not barking.
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. (More than likely because someone thought "Hey! They would never allow cameras in NORAD!")
Audio problem: When Jennifer gives David a ride home near the beginning of the film, as David gets on her bike, she looks round and tells him to "Hop on" without moving her lips.
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.
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.
Continuity mistake: The very first time when David connects to NORAD from his room and Jennifer is in the room with him she is wearing very short, shorts, but after they flash to the NORAD scene during the first simulated attack and we go back to David's room, Jennifer is now wearing long sweat pants.
Factual error: In the scene in which Prof. Falken shows David and Jennifer a dinosaur movie projected on a screen, Falken is between the projector and the screen, with portions of the image on his face, slightly blurry, just like it would be. But David and Jennifer aren't in the light path, so the only light on them would be reflected from the entire screen, but when they're shown, images from the dinosaur movie are perfectly focused on them, which would be impossible.
Plot hole: The whole point of playing tic-tac-toe in the end was so Joshua could learn that the game would always end a tie. But take a close look at the second game he plays in the war room. Two "X"s and two "O"s are played in boxes where all Joshua would have to do is put the next X in the top middle box, and he would've won that game. A human could conceivably miss that, but since it was just established that Joshua was playing himself (of players "zero"), a computer wouldn't have missed such an easy move, especially since the reason Joshua plays anything is "to win the game".
Factual error: When Matthew Broderick's character sets up his computer to look for other computers by getting it to dial a number, check to see if it is a computer, record the number if it is, hang up, and repeat the sequence, the writers forgot one thing. With the type of modem he was using, someone would have to physically push the hangup button on the phone before it could dial another number. Also, those types of modems only made the sounds to speak to another computer, they didn't make the dial tones needed to dial the next number. The next generation of modems that hooked directly to the phone line could do all of the above, but with what the movie used, it was impossible.
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.
Other mistake: While it makes for a great story for the movie, the reality is that all computer systems used by government agencies - especially in the DoD - are required to provide their source code for thorough inspection to ensure there are no back doors, hidden subroutines or other types of software code that are not relevant to the task designed for. Given the critical nature of this system being in control of launching nuclear missiles, the reality is that none of the games that Falken wrote - much less the whole routine to allow it to simulate a game like this - would still be in, for obvious reasons. Of course it could be argued Falken hid this, however given the lack of complexity for a computer of this era, it's highly unlikely.