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.
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.
Factual error: When Matthew Broderick 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 (a so-called "acoustic coupler"), someone would have to physically push the hangup button on the phone before it could dial another number. It could be entirely possible though to dial a number from the computer and send the acoustic coupler the instruction to generate DTMF tones, but this could only be done once; without an actual relay that will cut the line and get a new signal tone again, no looping (war dialing) could be done. 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.
Factual error: The beginning starts out with two Strategic Air Command officers in a Minuteman Intercontinental Ballistic Missile (ICBM) Launch Control Facility (LCF) going through the strict procedures for launch of an ICBM. The crypto messages are being received by the LCF, the launch officers pull the launch sequence and confirm the crypto launch command (encoded message) matches that which is contained in the launch sequence - all very secret, and all very realistic. However, the missile they show in the movie powering up for launch is a Titan ICBM, and how you launch a Titan from a Minuteman LCF is beyond me.
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. (00:07:25 - 00:08:25)
Factual error: 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.
Plot hole: Given the number of possible combinations the launch code could be (over 3600 trillion possibilities) it makes no sense that W.O.P.R cannot process that significantly quicker given how easily it's able to calculate thousands of ICBM impacts, damage inflicted, casualties, etc. for each "War Scenario" at the end as quickly as it does to determine a winner.
Plot hole: Why is David the only one aware that the computer is still playing the game? The NORAD people should have been aware the "simulation" was still running including the countdown. They believed the Soviets were really carrying out troop movements and bomber attacks over Alaska, did they never once look at the screen telling them "game time elapsed/remaining" and conclude none of it was real?
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.
Factual error: When Joshua/WOPR calls David back, the sound David hears on the phone not only isn't what modems actually sounded like, but in reality, the answering modem makes the initial sound to start communications, not the calling modem. The calling modem remains quiet until it hears what start tone the answering side answers with so it can start negotiating baud rate, encoding method (FSK vs PSK), etc.
Other mistake: When David and his dad are sitting at the dining table, dad butters his bread to roll his corn on the cob. You can see that the butter melts on the corn and bread, but then he bites into the corn and finds that it is raw. Raw means uncooked and the cob would be cold, therefore the butter wouldn't melt.