Factual error: When Riggs and Murtaugh go to investigate the scene where the body of Ramon Alvarez is discovered Scorsese states that Alvarez's dog tags say "SEAL" on them. His dog tags would not say that he is a SEAL on them. The information listed on dog tags is as follows: Name, Social Security Number, Blood Type, Service Branch, and Religious Preference. There is no way to tell what a given service member's MOS (job) from his or her dog tags.
Continuity mistake: When Riggs is hanging from the tree, his hands are tied behind his back. When he falls and lands on his back, his hands are still behind his back. In the next shot of him, his hands are now tied in front of his body, even though he didn't have time to do anything to get his hands there.
Factual error: Riggs and Murtaugh are told that Cooper's self defence argument (that Woodcock's pistol fired the fatal shot as they fought) holds up because Cooper's gun is 9mm, whereas the slug pulled from the body was a .358. There is no such thing as a handgun chambered in .358. The only .358 caliber that exists is .358 Winchester, which is a rifle round.
Continuity mistake: At the barber shop Murtaugh's son takes his headphones out to listen to his dad while he's talking to him. Right after there is a quick camera angle change and you can see in the mirror his son has his left headphone back in his ear. The headphone appears and disappears throughout the scene.