Continuity mistake: Just before Colonel Kurtz falls to the ground we catch a glimpse of him with blood running from the top of his head down the right side of his face. Several seconds later as the lies on the ground dying there are only a few small drops of blood on the right side of his face. (03:08:05)
Continuity mistake: The paint on Captain Willard's face changes many times throughout the movie. Two very noticeable examples are right after he kills the colonel, where the paint is almost entirely off his face, and several shots later as he leaves the temple heading back to the boat where his entire face is painted a much darker green then before. (03:09:20 - 03:11:50)
Continuity mistake: Captain Willard walks over to Colonel Kurtz' typewriter and sees his manuscript laying there. As he first looks down at the manuscript it's totally in the dark but in the next shot as he thumbs through it it's in the light. (03:10:55 - 03:13:20)
Continuity mistake: Captain Willard sits down at Colonel Kurtz' typewriter after he reads the manuscript and as he moves into the light we see that his right shoulder is covered with mud. To shots later as he emerges from the temple the mud is not there. (03:11:05)
Continuity mistake: After Captain Willard emerges from the room from killing Colonel Kurtz, he has a great deal of blood and oil on his chest and shoulders. When he grabs Lance to go back to the PBR he has mud and water droplets on his chest. (03:13:20)
Continuity mistake: As the PBR backs out of Kurtz' compound the person at the controls is wearing a shirt. Lance is sitting on the front of the boat so the only other person that could be at the helm is Captain Willard who is not wearing a shirt in the previous or next shots. (03:14:00)
Continuity mistake: The PBR's roof canvas is damaged by fire after passing another PBR. Later, Chef and Lance repair it with palm fronds. When departing the Do Loung bridge, the roof canvas appears undamaged.
Continuity mistake: In the wide shot when Kilgore throws the megaphone in the air all the smoke canisters that throughout the scene were blowing behind him are now gone.
Continuity mistake: When Captain Willard runs into the photographer when he reaches Kutz's compound the scene keeps jumping back and fourth between the two characters as they walk around. The photographer is wearing a red headband. As the film shots keep jumping back and fourth between the two characters the headband changes from being folded flat to being rolled up across his brow.
Continuity mistake: After Captain Willard smashes the mirror, there is no blood on his right hand and chest. 5 seconds later, there is.
Continuity mistake: When Colonel Kilgore is telling Lance about surfing in the front of the water, the sun changes between shots.
Continuity mistake: Kilgore is talking about how he wants to surf the beach. The lighting changes significantly in the space of seconds from the shots of the surf to the shots of the fighting. The shots of the surf are later in the day while the angle of the sun in the fighting shots is higher indicating afternoon.
Continuity mistake: When Willard has a severed head dropped on his lap, the stump is clean and flat like it was sliced by a samurai sword. When the head falls to the ground, the stump is bumpy and ragged like it was hacked with an ax.
Continuity mistake: Close to arriving to the French plantage, and also when leaving it, Captain Willards' helmet cover changes between a plain olive green and a camouflage.
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