Continuity mistake: When the soldiers come to the door with Captain Willard's new orders a green piece of paper (money) appears behind his gun lying on the table under the lamp. It's there in the close-up but not when they first walk in. The other pieces of paper in front and to the left of the gun are also moved. (00:08:20)
Continuity mistake: As Colonel Kurtz sits and reads articles to Captain Willard from Time magazine look very closely at the clasp of his dog tag necklace on the left of his neck. Throughout the shots he keeps both hands on the articles that he's reading yet the clasp on his necklace moves down his neck more than an inch without being touched. (02:50:55)
Continuity mistake: As Lance prepares Chief Phillips' body for burial at sea his left foot is on the side of the boat. In the next shot of Chef talking to Captain Willard, Lance is already in the water. (02:23:55)
Continuity mistake: Before the napalm attack we see a shot of the spotter plane flying over Vin Drin Dop and the place hasn't been attacked yet even though the battle has been going on for a while. There's no smoke and the bridge is still intact. (00:47:25)
Continuity mistake: Captain Willard reads Kurtz' file and narrates, "Late Summer, Autumn 1968" as Clean drums in front of him. At first Clean has both sticks on the helmet than in the next shot the right stick is on the helmet and the left is hitting the boat. (01:17:20)
Continuity mistake: At the dinner table General Corman tells Harrison Ford to, "Play that tape for the captain please" and we see a close-up of his watch and it reads 4:37. It's late for lunch but when Harrison Ford brought the can of Budweiser to the general it was lunchtime as his watch read 12:35. (00:12:35)
Continuity mistake: Dennis Hopper speaks a little too loud to Captain Willard while Colonel Kurtz is reading his poetry and he gets mad and throws a bowl of bananas right at them. In the next shot they both duck but nothing comes flying their way, no bowl and no bananas. (02:56:20)
Continuity mistake: Approaching the sampan Chief Phillips says, "These boats are running supplies in this Delta," and Captain Willard is holding Colonel Kurtz' folder it is left hand. In the next shot he's not holding anything in his left hand and using it to lean on the boat. (01:34:35)
Continuity mistake: As they all sit on the boat and read their letters from home Lance dances around with the purple smoke canister. Look closely as the red Vietnamese banner on the shoreline disappears between shots. (01:51:20)
Continuity mistake: The guy on the second patrol boat that throws the flare onto their roof starts out wearing a dark green beret, but in the next shot, when we see him up close, he's not wearing any hat at all. (01:16:30)
Continuity mistake: During dinner Hubert de Marais starts to rattle off all of the places in the world the French have lost wars. In the next shot the wine glass in front of Captain Willard goes from one quarter full to three quarters full between shots. (02:05:30)
Continuity mistake: Captain Willard tries to stop him from shooting but Chief Phillips yells, "you son of a bitch" as he pushes the captain to the floor. In the next shot the captain is up and holding onto the roof of the boat with his left hand. (02:22:25)
Continuity mistake: In some shots of the helicopters as they approach Vin Din Dop they're above the clouds, but in successive shots they're at or near ground level. (00:38:35)
Continuity mistake: Right after the napalm explosion there's a wide shot of the beach as Colonel Kilgore and the two soldiers with him look across at the destruction. We also see Lance sitting alone in the crater and Captain Willard is nowhere near them. They cut away to the PBR for three seconds then back to the crater and the Captain is sitting next to Lance again. (00:48:55)
Continuity mistake: As they start to eat with General Corman there's a hat hanging on a cabinet behind and to the right of the general and both doors are closed behind it. The next time we see it the right door of the cabinet has been opened but no one went near it. (00:11:45)
Continuity mistake: When Kilgore's chopper lands, Captain Willard is holding his rifle with the strap around his left shoulder so that the butt comes up above his head. In the next shot he's holding it in his left hand and the strap is not around his shoulder anymore. (00:26:40)
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: As they approach Kurtz' compound Captain Willard climbs up to the top of the boat. As he's climbing there's no helmet hanging off the light pole in front but in the next shot as he reaches the top a helmet appears. (02:27:40)
Continuity mistake: As "Satisfaction" plays in the background Clean dances behind Chief Phillips as he controls the boat. In the next long shot of Lance water skiing past the Vietnamese on the shoreline Clean is not there. (01:14:45)
Continuity mistake: As Captain Willard sits in the boat looking at pictures of Colonel Kurtz the shadow of a metal bar slowly goes from left to right across the pictures indicating that the boat is turning. In the next shot of Clean dancing to Satisfaction we can tell by looking at the sky that the boat is not turning but going straight ahead. (01:15:15)
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.