Here's a few more that come to mind:

Lou and Peter Berryman's "DO YOU THINK IT'S GONNA RAIN?" (which is also a cumulative song).

Lou and Peter Berryman's "If (Dueling Paranoias)."

"TWO SLEEPY PEOPLE," by Hoagy Carmichael and Frank Loesser, sung by Bob Hope and Shirley Ross in "THANKS FOR THE MEMORY," 1938.

And here's a related thread, although not all the songs are dialogues: DUETS, (And Other Multi-Part Songs).