Kilgore: I love the smell of napalm in the morning.
Willard: Hey soldier, do you know who's in command here?
Soldier: Ain't you?
Chef: Why do all you guys sit on your helmets?
Soldier: So we don't get our balls blown off.
Kilgore: What the hell do you know about surfing, Major? You're from goddamned New Jersey!
Willard: Everyone gets everything he wants. I wanted a mission, and for my sins, they gave me one. Brought it up to me like room service. It was a real choice mission, and when it was over, I never wanted another.
Kilgore: Charlie don't surf!
Kilgore: You either surf or you fight.
Lance: Disneyland. Fuck, man, this is better than Disneyland.
Willard: If that's how Kilgore fought the war I began to wonder what they really had against Kurtz. It wasn't just insanity and murder, there was enough of that to go around for everyone.
Kurtz: We must kill them. We must incinerate them. Pig after pig. Cow after cow. Village after village. Army after army.
Kurtz: The horror... The horror.
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