Deliberate mistake: None of the helicopter's in the movie have the Army markings seen on Vietnam era Hueys. The most you'll see is some crooked, faint and inaccurate stencils in the wrong spots. (00:37:50 - 00:51:25)
Deliberate mistake: Martin Sheen has a cut on his cheek that starts as a scar and becomes an open wound. (00:02:27)
Deliberate mistake: After Hubert exit's the French Plantation dinner scene the stage lighting on Roxanne and Captain Willard shuts down abruptly before their close-ups. There's no one there to touch a light switch and we can see outside and the natural lighting hasn't changed. (02:13:30)
Chosen answer: Kurtz was protected by political connections and his military record from being relieved of command, and the Montagnard people whose village he was in were considered American allies, so they couldn't just send a force in to kill him or relieve him of command, but an individual assassin. (The US Government's "official" policy is that they do not engage in assassination, which is why "this mission does not exist, nor will it ever exist").
Captain Defenestrator