Factual error: In the scene where Riggs and Murtaugh are in the police car, they intercept a transmission in Afrikaans for the first time in the movie. Except it's definitely not Afrikaans (my home language). I think it's Vlaams or Dutch.
Plot hole: Riggs and Murtagh send the car flying out of the container, pushing the money out as it goes. Ignoring that the car couldn't possibly force the sealed container doors open, even without having the obstacle of pallets loaded with money in the way, the car actually explodes in mid air for no reason. (Director's Cut.) (01:44:35)
Revealing mistake: Movie physics occasionally dictate random explosions, but this is too bizarre. Near the start, a cop car smashes through a caravan, and the caravan blows up. Fair enough, may have hit a gas cylinder or whatever. We see the cop car, practically undamaged, skid to a halt. Cut to a few reaction shots, then suddenly the cop car just blows up for absolutely no reason whatsoever. (00:02:25)
Plot hole: The hitman in the tow truck leans into the rear/left corner of the cab as the surfboard approaches. The board hits the glass at the very top, dead center. Hence, it definitely misses him completely, yet the hitman is killed. Windshields don't break into "shards" (which is another mistake made listed here, and as it can't exist, it can't be the cause of death) and the roof is not damaged. He is the driver, and he stops the truck by braking. There is no vehicle collision. It is clear to all viewers that the makers intended the surfboard to be the cause of death - nobody watching the film thought otherwise.
Factual error: The amount of money that leaves the cargo container when Riggs and Murtaugh use the car as a battering ram is nowhere near what was inside considering there were at least 4 or so pallets in front of the car. The car didn't have anywhere near the momentum to drive 'through' the pallets and leave them inside.
Continuity mistake: In the scene where Murtaugh is shooting his attackers with the nail gun the walls are apparently covered with super plastic. How is it that the plastic sheeting is light enough for the bad guy to punch through, but is able to support the weight of his corpse by a single nail when Murtaugh shoots him through the plastic? Wouldn't the plastic tear and drop the body to the floor? (01:15:45)
Continuity mistake: Near the end, when Riggs is captured by the bad guys they tie him in a straight-jacket and toss him off a pier into the drink, he's wearing a pair of Dingo style boots. Underwater when he dislocates his shoulder and escapes the straight-jacket, he sees Patsy Kendit's corpse, swims over, checks her pulse and swims to the surface. Now he's barefoot. Yet when he attacks and kills the two thugs he's got the boots on again. They can even be heard on the pavement.
Revealing mistake: When the small car is launched into the air, the underside is plainly visible, showing no normal working parts like a suspension or drivetrain. It has a flat pan bottom with a square-tube support structure to make it very light for its flight and to retain it shape. (Director's Cut.) (00:42:40)
Factual error: Nail guns like the one used as a weapon here can not be fired unless the tip is pressed against a hard surface. It will not fire a nail at someone across any space. The first guy is killed by pressing the gun to his forehead, but the second is hit from several inches away (evidenced by the length of each nail still sticking out of the guy instead of them being fully into his body with only the nail heads showing.
Factual error: In the scene where the car blasts out the container, the weight of those stacks of money would make it impossible, for a car parked so close to them, to blast them out the way like it does. There must be around 18 pallets of money stacked up on top of each other, even if they only weighed around 500kg each, that's 9000kg.
Continuity mistake: During a chase scene involving Murtaugh's station wagon and a flatbed tow truck, the hotel waiter shoots at Riggs through the truck's rear window. In this series of shots, notice how Riggs is swinging by a chrome handle on the passenger's side - which wasn't there during the gunpoint hijack shot just moments ago. (00:39:30 - 00:40:45)