Visible crew/equipment: After Chief Phillips is speared if you look closely at his left shoulder you can see his shirt bunched up by the harness underneath that holds both ends of the spear to his body. Later you can see a blood bag underneath his tee shirt as he lies on the floor of the boat dying. (02:22:40)
Continuity mistake: After she sits down everyone is eating and Roxanne breaks the silence saying something in French while holding her napkin. At first she holds the napkin with the frills up but in the next shot the napkin is reversed with the frills on the bottom. (02:04:25)
Continuity mistake: As they listen to the Frenchmen on the plantation dock, Chef picks up his rifle from the side of the boat. Note this spot and the bloody footprints there, which appear and disappear between shots. (01:57:40)
Visible crew/equipment: As they pull away from the Do Lung Bridge, look closely and you can see water spraying up and turbulence from the propellers of the camera boat in front of them. (01:48:40)
Continuity mistake: When Colonel Kilgore shows Lance the surf as they fly over the beach he says, "look, good six-foot swells" and his surf board is not on the side of his helicopter as it is in the next shot when they land. (00:45:10)
Continuity mistake: Colonel Kilgore walks over to a VC soldier with a stomach wound and one of his soldiers follows behind carrying a thermos with an orange cap in his right hand and nothing in his left. Kilgore turns and says to this soldier, "Give me that canteen" and in the next shot the canteen appears in his left hand. (00:29:20)
Continuity mistake: Colonel Kurtz sits at the door to the dark room where Captain Willard is being held reading articles to him from Time magazine, and it's obvious that the light is coming from the door in front of him. The door doesn't close nor does the captain stop looking forward, but the lighting changes throughout the scene. In some of the close-up shots his entire face and neck are lit up and in other shots shadows fall across his face and neck. (02:51:15)
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)
Continuity mistake: As Chef and Captain Willard walk through the jungle we get a couple of good close-ups of the band-aid on his cheek and it's different between shots. In the close-up when he first hears the tiger, the right side of the band-aid is pointing towards the top of his ear lobe. Then just seconds before the tiger attacks we see it pointing directly into the middle of his ear. (00:57:50 - 00:58:45)
Continuity mistake: Dennis Hopper says, "Zap 'em with your siren, man. Zap 'em with your siren," as they get close to shore. At first there's no smoke and unarmed natives behind him and the smoke changes to red in the close-up. Then in the next close up there's orange smoke and armed troops behind him. (02:29:15)
Continuity mistake: Roxanne continues to prepare the pipe for Captain Willard and says, "he would rage and he would cry, my lost soldier, and I said to him there are two of you". In the shot of them from the foot of the bed Roxanne's head is almost touching the captain's but in the next shot over the side of the bed she's about a foot away. (02:16:55)
Continuity mistake: Chief Phillips Orders Clean down onto the sampan to help Chef with the injured Vietnamese woman. We hear Clean talking to Chef as they lift her body but the next time we see the Vietnamese woman Chef is the only one there and Clean is back up at his machine gun. (01:38:05)
Continuity mistake: Just before the Medevac helicopter lands in the Vin Drin Dop plaza, a soldier machine guns 13 urns sitting next to the building on the left of the courtyard and the building also explodes. In the next shot of the medics working on the injured soldier the 13 urns are back and the building hasn't exploded yet. (00:43:05)
Continuity mistake: Before Captain Willard shoots the injured Vietnamese woman on the sampan he's holding onto a metal bar with his left hand. In the next shot he's not holding onto anything. (01:38:25)
Continuity mistake: During the beach party as Captain Willard is saying, "He was one of those guys that had that weird light around him." Note he is holding a cigarette and drinking a Schlitz, with four cans of Budweiser in front of him. In his next shot, just seconds later, there is another can of beer and the cigarette is gone despite the fact he has not moved at all. (00:32:00)
Continuity mistake: At dinner with the general, Harrison Ford walks over to the captain with the picture of Colonel Kurtz and he has a pencil in his right hand. In the next shot as he drops the folder on the floor the pencil is not there. (00:12:25)
Continuity mistake: In the beginning of the movie when Captain Willard hears the helicopter flying over and walks to his window the blinds are dark. In the next shot when you see him looking through them they're lit up. (00:04:15)
Continuity mistake: Dennis Hopper stands on the back of the boat and says to Captain Willard, "Yeah, well, they think you've come to uh - , to take him away". If you look to the left of the screen you'll see a piece of seaweed hanging off the roof of the boat. When Dennis Hopper walks back to the front of boat it's gone. (02:30:15)
Continuity mistake: When they first approach Dennis Hopper we see the water on the left side of the boat past where Lance is sitting and there are no flames on it. After Hopper jumps on the boat and goes to shake the captain's hand we see an oil slick burning in the middle of the water. (02:29:40)
Continuity mistake: In the shot of a priest saying the Lord's Prayer, we see smoke rising from an attack by a flamethrower behind him. In the next shot there's no smoke until the flamethrower fires again. (00:30:55)
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.