Said Auntie Ann, reacting to my implication that the lyrics — heard in the movie — were omitted from the TV show because they were about suicide. Actually, I don't know whether taking the words out was a kind of censorship or whether — more likely — they were rejected because they didn't set up the story for an ongoing series of episodes. It wasn't like "The Beverly Hillbillies" or "Green Acres," where the song fills you in on how these characters got into this situation.
And I'd be surprised if even 50% of audience for the "M*A*S*H" TV show knew the lyrics from the movie version. That TV show became much bigger than the movie. The movie came out in 1970, and the TV show was on from 1972 until 1983, back in the days when people didn't have VCRs, so it wasn't easy to go back and check out a movie you'd missed. But even if it had been easy, I think people became very attached to Alan Alda in the main role and wouldn't have enjoyed seeing Donald Sutherland horn in on it. And the main character in the movie is the Elliot Gould character, Trapper John. [OR: He was equally important.] These barriers are hard to cross. I loved the movie, and I didn't want to see the actors that I knew replaced by the warmer, fuzzier TV personalities.
On the subject of TV themes played without the lyrics, do you know the words to the theme whistled on "The Andy Griffith Show"? Here's Andy singing them. [ADDED: The words are about fishing. Wouldn't it be a kick in the head if it turned out that theme was also about suicide?]