TOD:
I read the other posts looking for ideas, but the only one I'd even heard of was The Lonely Goatherd. I do like that one. What else comes to mind is:
"Officer Krupke" from West Side Story
"Think of the Time I Save" from Pajama Game
"I Hate Men" from Kiss Me Kate
"Sounds While Selling" from She Loves Me
"Be a Dentist" from Little Shop of Horrors
"How Could You Believe Me?" from Royal Wedding
"Do You Love Me?" from Fiddler on the Roof
"I'm Not Getting Married Today" from Company
"Barcelona" from Company
I tried Googling this topic and apparently there's a lot of funny songs in Avenue Q, but I've never seen it.
I also remember "Thoroughly Modern Millie" and "The Drowsy Chaperone" being very funny shows, but I don't remember any specific songs.