Continuity mistake: During Willard's bout of insanity in the opening scene, we see him doing karate or something in front of his bed and there's blood visible on the sheets. Yet that is not supposed to be there until he smashes the mirror (half a minute later) with his hand and then runs the blood over the bed and his face.
Continuity mistake: When they first meet, Colonel Kurtz says to Captain Willard, "Have you ever considered any real freedoms?" and the Captain's shadow appears on the stone column to the right of the Colonel. The Captain is kneeling down and hasn't moved from his spot, but when we saw this column before his shadow was not anywhere near it.
Continuity mistake: Before they board the sampan Lance says, "It's clear, Chief," and we see a wide shot of the sampan and PBR. No one goes near it, but for just one shot the dress on the clothesline is rearranged. The two straps it hangs from are pushed together. In all the other shots the straps are wide apart.
Continuity mistake: Harrison Ford lights up his cigarette and Captain Willard says, "I'm not presently disposed to discuss those operations sir." Jerry is sitting with his back almost up against the wall. In the next shot as he takes a puff from his cigarette he's moved and is now leaning forward.
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.