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: 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.