Deliberate mistake: The way diplomatic immunity is depicted in the movie is extremely inaccurate. While technically a sufficiently ranking diplomat could perform the acts shown in the movie without being subject to arrest, their home government would be requested to turn the diplomats over for trial, or the diplomat would be declared persona non grata and expelled from the country. A drug smuggling ring and murdering multiple policemen is far, far more serious than actual crimes that diplomats have been expelled (or handed over for trial) for in the real world. Additionally, the consul general and his officers would not be considered diplomats. They would only have limited consular immunity with regards to "official acts" in the course of consular business.
Audio problem: In the scene where Leo explains how he launders the drug money and Roger says "You're a cheat", Leo says "Come on, everyone cheats a little, look at the pentagon." But when he says "look at the pentagon", it's dubbed in because the sound of his voice doesn't match his other dialogue, and if you read his lips it doesn't match either.
Plot hole: During Riggs' last fight with Adolph, Adolph holds the knife in his own hand as Riggs forces him to stab himself very slowly. At any point, Adolph could simply have let the knife fall or just tilt down before that happened. Nothing Riggs is doing is making such impossible. "Locking" his wrist does not prevent him from letting go of the knife. It only prevents him flexing his wrist. And this guy is a trained killer, fighting for his life. Better to battle for control of the knife than just let it kill you.
Continuity mistake: Near the end of the car chase you see a shot of the station wagon Riggs and Murtaugh are riding in get held up by a couple of police cars. The hood is popped open. The shot changes to the BMW crashing into the store window and then back to the station wagon. The hood is down.
Continuity mistake: When the bathroom explodes, from one angle we see the toilet come sailing out over the roof of a police truck parked at the front of the house. While the angles make it a bit tricky to judge, there's a definite leftward trajectory. We then cut to a closeup of Murtaugh's car, and the toilet lands on it as if it was flying to the right. Even if the angles are misleading, in the first shot the police truck and other cars are parked to the right of the exploding window. In the shot of Roger's car he's also parked to the right, but without all the cars visible around it which there should be.
Continuity mistake: In the scene where Riggs and Rika are in the trailer while the men in the helicopters are shooting, after the scene where you see the T.V. fall down, You see the cabinet being shot. If you look carefully, you can see pieces of the cabinet fall down, then appear again. Director's cut.
Continuity mistake: When Hans first walks into the room with the fishtank, the plastic on the floor flaps up behind him as he walks, ending up folded up on itself without much space between the man and the end of the plastic. When he gets shot and falls back onto it, it's perfectly flat and he's a lot further forward.