Continuity mistake: As Clean says, "I gotta tape from my mom," we see a rope tied around the cleat on the starboard side of the boat. He doesn't go near the cleat as he listens to the tape but the next time we see this spot, right when the attacks starts, the rope is gone. (01:51:00)
Continuity mistake: At the beach party LColonel Kilgore stands up and says, "This is the first of the ninth, Air-Cav, son". If you look closely when he stands up the map that they were just looking at is gone. The colonel was holding it, but he doesn't have it nor does Captain Willard. (00:33:40)
Continuity mistake: As they sit in the boat at Kurtz's compound discussing what they should do, Captain Willard tells Chef about the air strike and hands the radio to him with the antenna standing straight up. In the next shot as he grabs it the antenna is folded over. (02:35:45)
Continuity mistake: As the camera pans along from the rear of the helicopter to the front Chef says to Miss May, "I've got every one of your pictures". Watch very closely as the camera shoots through the two windows on the helicopter's door. The Styrofoam head with the black wig on it that Chef asks the playmate to put on is not up against the left window as it is several shots later inside the helicopter. (01:27:45)
Continuity mistake: Roxanne prepares a pipe for Captain Willard and says, "It was the morphine he took for the wounds he suffered in his heart." The shadow of her left hand falls on her face as she holds the needle over the flame. In the next shot, however, the lighting has changed and the shadow is now nowhere near her face. (02:16:40)
Continuity mistake: On the boat Captain Willard reads newspaper clippings about Colonel Kurtz' and thinks to himself, "38 fucking years old." He is sweating profusely, with sweat dripping off of his face and neck. However, his right hand is perfectly dry when he lifts a flashlight and moments later a cigarette. (01:01:00 - 01:02:50)
Continuity mistake: Colonel Kurtz leaves after he finishes reading the Time articles and there's a close-up of Captain Willard's face and chest. He's soaking wet in the close-up but in the next shot, as he collapses trying to leave the room, his face and chest are dry and there isn't as much dirt on the left side of his chest. (02:52:25)
Continuity mistake: The gentleman sitting next to Roxanne in the tan jacket gets up and says, "Look, this is the truth, an egg, the white left, but the yellow stays". If you look closely the lazy Susan in the center of the table holding the oil and vinegar has been rotated between shots without being touched by anyone. (02:07:20)
Continuity mistake: The large bell shaped radar antenna is missing in several of the long shots of the boat as they go up the Nung River. It was there when the roof caught fire and we see it back again when they meet up with the Playboy bunnies (redux version). (01:20:30)
Continuity mistake: As the chopper pilot circles the courtyard he says, "We've got wounded down there" and a yellow flair is in the center of the courtyard. In the next shot of the soldiers running around down below we see the center of the courtyard but the yellow flare is not there having been replaced by a Vietnamese hat. (00:43:10)
Continuity mistake: The old gentleman sitting across the table from Hubert de Marais says, "Maybe in the future we can make something with the Vietminh" and the gentleman to the left of Roxanne replies, "Don't you understand". If you look closely the cigar that he just put into the ashtray in the previous shot, along with its smoke, has disappeared. (02:07:00)
Continuity mistake: At the Playboy Bunnies camps Lance picks up the second barrel of diesel fuel with both hands as he takes it off the boat. In the next shot both his arms are wrapped around it. (01:27:20)
Continuity mistake: After the helicopters land on the beach at Vin Drin Dop a soldier yells, "I'm not going, I'm not going" and we see huts that were blown up and burned down earlier in the scene untouched and as good as new. (00:42:50)
Continuity mistake: At the beginning of the arrow attack Chief Phillips yells, "Chef, open up, fire". Before he says this we see a wide shot of the shoreline with no vertical poles sticking up. In the next close-up shot of Chief Phillips there's a red spear stuck in the mud on the shore. (02:21:35)
Continuity mistake: Near the beginning of the movie when we first see Chef's tattoos up close the tip of the serpent's tongue on his left arm is entirely red. Later in the movie before he gets onto the Vietnamese sampan we see it again up close and the tip of the serpent's tongue is green. (00:24:05 - 01:35:55)
Continuity mistake: Chef and Clean pull Lance entirely into the boat. Than Captain Willard says, "Don't leave without me "as he runs to steal the colonel's surf board and we see Lance's legs dangling over the side as he is pulled into the boat a second time. (00:51:20)
Continuity mistake: After he cuts his hand on the mirror Captain Willard rubs his face with his bloody right hand. In the next shot as he attempts to pour a drink into his mouth only his thumb is bloody and there is no blood on his face. (00:07:10)
Continuity mistake: After the gentleman in the tan jacket spills the egg into the plate, between shots Roxanne goes from having her right hand on the table to having her right hand near her mouth holding her cigar. (02:07:30)
Continuity mistake: Harrison Ford goes over to the tape recorder to play a tape of Colonel Kurtz' voice and the food on the captain's plate changes completely. At first there's a serving of roast beef with a roll next to it, than when we see it again there's less roast beef and pees and carrots where the roll was. (00:12:55)
Continuity mistake: At the funeral Chief Phillips picks up the flag from Clean's body holding it between his fingers about two inches in on each end. In the next shot he's holding it on the edges and the flag is reversed with the stars and stripes on the right. (02:01:15)
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.