Continuity mistake: The Vietnamese woman on the sampan has her hair in two ponytails when Chef shoves her. It's in disarray when she falls then in one ponytail as she is shot by Clean. (01:36:25)
Continuity mistake: The Frenchmen with the accordion asks Captain Willard, "Do you know anything about Dien Bien Phu?". If you look closely at the doors in the other room, most of the lower shutters are closed. The next time we see them all as the Frenchman playing the accordion stands behind the captain and says, "Dien Bien Phu, okay" all the shutters are open even though no one goes near them. (02:08:30 - 02:10:00)
Continuity mistake: When Chief Phillips is speared we first see Lance at the helm of the boat and it's going forward. In the next shot after Chief Phillips is speared the boat is going around in circles. (02:22:30)
Audio problem: On the French plantation dock Lance picks up what looks like the sole of an old shoe and smells it. If you look closely you hear a sniffing sound that isn't in sync with the movement of his chest because he doesn't really inhale while that horrible thing is near his nose. (01:57:05)
Continuity mistake: Lieutenant Carlsen says, "Get that light off me" as the PBR approaches the shore. Lance, sitting at the front machine gun, starts to move the spotlight before the lieutenant says anything then in the next shot the spotlight is already facing him. (01:41:40)
Continuity mistake: Hubert has a discussion with the Frenchmen sitting to the right of Roxanne who gets up and leaves. As he gets up his hands are on table but in the next shot they're at his sides. (02:11:05)
Continuity mistake: Chef and Clean pull Lance into the boat after they ditch Col. Kilgore but in the next shot as Lance climbs in the boat Clean has disappeared. (00:51:15)
Continuity mistake: Captain Willard sits in the sunshine near the back of the boat reading his file on Colonel Kurtz but in the wide shots of the boat as Lance is water skiing he's not there. (01:15:20)
Continuity mistake: After Harrison Ford lights up his cigarette and walks towards the General, we see Jerry sitting behind him putting the piece of paper he was reading into his briefcase. In the next shot Jerry is holding the piece of paper in his left hand again. (00:10:40)
Continuity mistake: As Captain Willard sits in the boat reading and thinking to himself about Colonel Kurtz he unwraps and eats a Hershey bar. He crumples the wrapper with both hands and throws it out of the boat, leaving nothing in his hands. In the next shot he's eating the Hershey bar again and the wrapper is back. (01:15:45)
Continuity mistake: After Willard kills Kurtz, and Willard is leafing through some papers that belonged to Kurtz, he opens the memoirs halfway through the stack. As he closes the stack, the visible page changes to a different page and different placement. (02:22:00)
Continuity mistake: Colonel Kilgore talks about getting the boat up the river during the beach party and says, "Hell, a six foot peak," and a cigarette appears in his mouth between shots. (00:33:50)
Continuity mistake: When they leave there are two fires on the Do Lung Bridge as Clean looks on manning the rear machine gun. In the next shot there is only one fire on the bridge. (01:48:45)
Continuity mistake: After he hears the Frenchmen shouting at them Chef turns around and looks at Chief Phillips as he puts his gun down on the side of the boat. It in the next shot he's looking in the opposite direction. (01:57:30)
Continuity mistake: The next day as the boat leaves the USO show at Hau Phat to continue up the river we see the battered stage from the back of the boat. The boat passes the stage by about 100 feet but in the next shot its back right next to the stage. (01:11:25)
Continuity mistake: After Captain Willard looks at the picture of Colonel Kurtz being decorated by General Westmorland, they're all startled by the "Arc light" B-52 strike, note the rear machine gun is pointing forward and a helmet is hanging from the middle machine gun. In the next shot the rear machine gun is pointing aft and the helmet is now gone. (00:24:05)
Continuity mistake: The actual napalm attack burns quite a large area of the tree line and just two shots later Colonel Kilgore speaks the famous lines, "I love the smell of napalm in the morning!" At this time, when we see this area is still smoking, quite heavily, however, the amount of destruction is much less and there is very little fire visible. (00:48:55)
Continuity mistake: On the dock Chef listens intently to the Frenchmen who is yelling at them. A smoke canister is lit up on the back of the boat but in the next shot there's a close-up of Chief Phillips with no smoke behind him. (01:57:25)
Continuity mistake: As they prepare to lower Clean's body into his grave Lance stands to his right and reaches down to lift the flag. According to his wrist watch it's 11:45 but twenty seconds later there's a close-up of Chief Phillips' watch and it's now 4:10. (02:01:10)
Factual error: When LtC. Kilgore requests the air strike, you can see that the F-5s are not carrying any ordnance under their wings, except for a centerline fuel tank. Also, their color scheme varies from a white to a greyish one. (00:47:48)
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.