Character mistake: When Colonel Kilgore is handing out death cards to the dead VC he says, "eight of spades". If you look closely at the card that he throws as it flies threw the air it's the six of spades. (00:28:00)
Character mistake: After Captain Willard walks through the front door of the burnt out French plantation dock there's a close-up of Chief Phillips at the wheel the boat. He turns around and looks right at Chef and says, "Lance". (01:56:40)
Character mistake: When Martin Sheen is on the boat, looking over Col. Kurtz's school papers, there is a dissertation by Kurtz on the history of the Philippines until 1905. The word Philippines is spelled "Phillipines"
Answer: It's not any specific song; it's just the kind of generic piano music you'd hear at dinner at a high-class restaurant of the era. Think of scenes in movies, films, etc. set or made in the '50s and '60s, where the characters go to a nice dinner and there's someone at the piano playing unobtrusive music to accompany the food/conversation.