I think the Tom Lehrer song you folks are thinking of is "I Hold Your Little Hand In Mine," isn't it. The old standard "Jalousie" ("Jealousy") goes back at least to the 1940s and probably the '30s, and the parody, like Uncle Dave says, probably came out shortly thereafter. [Could it have been a Spike Jones or Homer and Jethro version?]

So far, we've got:
"Leprosy, it's crawling all over me-
There goes my eyeball
Into my highball.
Oh My darling won't you
kiss me quick,
There goes my upper lip,
There goes my ear,
right into my beer!"

BTW, I always saw the song title printed as "Jalousie" on old records and sheet music. I thought that was the French spelling, though I've never looked it up. [Creole or Cajon, maybe?]