Continuity mistake: As the boat approaches Kurtz' compound the natives in the canoes are looking forward. In the next shot they're facing sideways looking at the PBR. (02:28:15)
Continuity mistake: As Lance prepares Chief Phillips' body for burial at sea they are both surrounded by orange smoke. In the previous shot the PBR was surrounded by purple smoke. (02:23:50)
Continuity mistake: Lance comes down from the roof of the boat and we get a good close-up look at his makeup. Fifteen seconds later as the arrows start flying his facial makeup is entirely different (neck, nose, ears, bandanna). (02:21:15)
Continuity mistake: As Captain Willard takes a smoke, the shadow of the pipe is on Roxanne's face but in the next shot as she looks at him the pipe's shadow has disappeared. (02:17:30)
Continuity mistake: As Captain Willard takes a smoke from Roxanne's pipe the pillow he's lying on is light green. Later after they're finished as he lies on his back the pillow is now light red. (02:17:20)
Deliberate mistake: After Hubert exit's the French Plantation dinner scene the stage lighting on Roxanne and Captain Willard shuts down abruptly before their close-ups. There's no one there to touch a light switch and we can see outside and the natural lighting hasn't changed. (02:13:30)
Continuity mistake: As they stand and listen to the Frenchmen shouting at them, the floodlight on the front of the boat, along with the machine gun turret that it's attached to, move about 45 degrees to the left between shots without having any one touch them. (01:57:35)
Continuity mistake: As Captain Willard jumps off the boat at the French plantation we see a wide shot of the debris on the dock. The fuel drums to the right of the screen are rearranged two shots later as he walks up to the door. (01:56:25)
Continuity mistake: As they approached the French plantation the partially crushed fuel drum on the back the PBR that we saw after Clean was shot, has been replaced with one that is only slightly dented. (01:56:15)
Other mistake: Clean gets some tapes from his mom back home but if you look closely at the back of his tape recorder as he grabs it there's no way for him to play the tapes without batteries in it. (01:51:00)
Continuity mistake: As they all read their mail from home Lance ignites a canister of purple smoke and it's sunset. In the next shot of Clean grabbing his tape recorder it's the afternoon and the sun isn't even shining. (01:50:55)
Continuity mistake: As they approach the Do Lung Bridge Chef talks to Lance with his gun muzzle pointing out in front of him. The muzzle isn't near anything but in the next shot it's resting on the flat piece of metal on top of the machine gun. (01:40:35)
Continuity mistake: As Chef looks over the papers from the sampan crew the woman is standing under the canopy with the sun shining on her. The next time we see her several seconds later she's in the same spot but the sun is no longer shining on her. (01:35:35)
Continuity mistake: Watching the USO show, Clean and Chef start out on the right of the stage with a still photographer and press reporters, who are sitting behind a desk. Later in a rear close-up of playmate Carrie Foster we see the entire right side of the stage but there is no press desk. (01:07:25 - 01:09:00)
Continuity mistake: As Captain Willard holds Colonel Kurtz' paper titled, "Commitment and Counterinsurgency" in front of the camera it changes color from yellow in the first shot to white in the second. (01:15:40)
Continuity mistake: As Satisfaction plays in the background, Lance is holding on to a white tow cord as he skis. Earlier as they pull out of Hau Phat we see him holding a blue tow cord before he unwraps his new water skis. Thirty seconds later in another shot he is holding the blue tow cord. (01:15:05)
Continuity mistake: As they approach the USO show at Hau Phat, Clean has a very large zit on the left of his chin. Later in this scene as he's speaking to the sergeant about getting some fuel it's gone. (01:03:30 - 01:04:55)
Other mistake: Captain Willard hears the drumming as they approach the USO show at Hau Phat. He's sweating profusely but they don't put any sweat on Chief Phillips, Lance or Clean who, in the same heat, are bone dry. (01:03:15)
Continuity mistake: As Captain Willard tries to get Lance away from Colonel Kilgore the band-aid on his right cheek angles upward as it nears his eye. Later we see it up close as they pull him into the boat with the surfboard and it does not angle upward but goes straight across his face. (00:50:40 - 00:51:45)
Continuity mistake: Before Colonel Kilgore looks at Captain Willard and says, "Someday this war's gonna end," the Vietnamese women and the soldier waving behind him to his left have moved between shots. (00:49:45)
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.