Other mistake: As Captain Willard gets up, he holds the Schlitz beer can upside down. (00:32:17)
Continuity mistake: Colonel Kilgore offers the wounded VC soldier a drink of water and his right elbow is leaning on his leg. In the next shot only his forearm is leaning on his leg. (00:29:45)
Continuity mistake: As Kilgore introduces Lance to his surfing buddies he says to him, " none of us are anywhere near your class though". Captain Willard's cigarette and it's ash gets shorter between shots. (00:30:25)
Continuity mistake: As they approach the shore to meet up with Colonel Kilgore, there's another boat with teeth painted on it's front next to them on their right. In the next shot they arrive on shore and the other boat is much further behind them. (00:25:00)
Continuity mistake: Captain Willard gets off the PBR and goes looking for Kilgore. A large piece of driftwood appears floating in the water to the right of the boat between shots. (00:25:15)
Continuity mistake: As they come to shore to meet up with Colonel Kilgore, the Captain is crouching on the front of the PBR. In the next shot as he jumps off the boat he's standing. (00:25:15)
Continuity mistake: As they all sit on the boat and listen to the "Arc light" B-52 strike Lance is sunbathing in direct sunlight. In the next shot he's in the shadow of the roof of the boat as he gets up and says "there they are". (00:24:15)
Continuity mistake: After the General says, "And very obviously he has reached his" there's a close-up of Captain Willard and his face is dry. The next time we see him a few seconds later his face especially around his chin is sweaty. (00:17:10)
Continuity mistake: As the general reminisces about Colonel Kurtz being a man of, "wit and humor" the roll on his plate moves and changes from being whole to having a bite out of it and back. (00:14:40)
Continuity mistake: As everyone sits down to have dinner at the General's all the glasses are empty. The next time we see them they're full but no one went around the table filling them. (00:11:40)
Continuity mistake: At General Corman's they sit down to have dinner and the ID card on his right chest pocket is attached to his shirt differently in various shots. At first it's not even on the pocket then later in the scene it's set right again. (00:11:40)
Continuity mistake: As he speaks to Captain Willard in the general's residence, Harrison Ford lights up a cigarette, leaving a big puff of smoke in the air behind him as he walks away. In the next shot the air is clear without a trace of smoke. (00:10:40)
Continuity mistake: After Clean is killed, the blood splatter on the PBR's metal plate changes. (01:52:34 - 01:52:58)
Continuity mistake: As Lieutenant General Corman walks to the table, all of the hutch cabinet doors are closed. 17 seconds later, the left end and right end hutch cabinet doors are open. (00:11:40)
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.