Plot hole: When the armored guy is propelled under the tanker trailer, his backpack explodes. That blast can not have been large enough to lift the entire trailer through the air. The trailer never explodes itself, and if it did, the tank itself would have torn apart and the frame beneath would have stayed on the ground, pushed down by that blast above. (Oh, and there's no such "safety feature" that is intended to launch a trailer in the air in case of an accident.) (00:05:00)
Continuity mistake: During the mobile home chase, you can see a white car following Murtaugh. In the next shot, when he starts to overtake the mobile home, the car is black.
Deliberate mistake: When Riggs is chasing the guy on foot, you can clearly see it is a stunt double, who is also very apparent in other shots throughout the film.
Factual error: When the sedan is struck by the train, it launches into the air and explodes, but doesn't sustain severe impact damage. The car rolls over in midair and it is nearly straight - not heavily bent or crushed as the entire left side should have been. Even a slow moving train heavily batters the passenger vehicle it hits. (00:17:05)
Factual error: Early in the film, while Riggs, Roger and Leo are on the Code 7, Riggs has to place his Beretta in his mouth so he can board the larger ship. The Beretta 92FS is a heavy handgun, it would be very difficult, if not impossible, for it to be held in the mouth, let alone to wave your head about like in the film. I own a Beretta 92 and I've tried it, so I'm speaking from experience.
Plot hole: When Riggs, Murtaugh, Lorna and the others are tied up and lying on the floor of the burning house, they can't stop coughing. However, the bad guys (who are also in the burning house) are just standing around laughing, wouldn't you think that they would be inhaling smoke also?
Continuity mistake: In the trailer chase scene, when Riggs is being dragged by the plastic wrap, it cuts away to a far shot of Riggs on the table. The "plastic" he is holding on to is very obviously a thick metal cable wrapped in saran wrap or something to make it look like plastic.
Revealing mistake: During the car chase when the Pontiac is about to jump into the office building, you can see a white stunt driver from the left side mirror, though Murtaugh is driving.
Other mistake: When Riggs and Murtaugh are assigned as new captains, they throw their old badges to the Chief and he catches them in his left hand as he grabs the new badges out of the desk draw with his right. In the next shot where he passes them the new badges, he is not shown to have swapped the badges over, so essentially he passed the old ones back to them.
Revealing mistake: In the scene where Riggs and Murtaugh's vehicle breaks through the glass to get back on highway, the vehicle sparks even before it hits the pavement.
Continuity mistake: In the scene where Riggs is at his wife's grave, Leo comes along to give him advice. In the scene he was in previously he had cuts across his face. At the cemetery the cuts are gone, then they reappear in the following scene at the hospital.