I think the telephone calls probably first showed up as a way to advance the plot(pre-cell phone) but they turned into a kind of subliminal gag. The telephone call Columbo receives in “Columbo Goes To College” is certainly staged since it gives Justin and Cooper information they use to incriminate themselves.
I agree with all the points so far...but in "Requiem," i actually got the impression that Columbo was annoyed with his department for calling at that point and letting Nora Chandler know too soon what department he was from.
When Nora says "which department?!" Columbo makes a slightly exasperated face, and when he gets on the phone with Homicide, he says, "...don't bother me now!"
Does anyone know of an episode where Columbo does NOT receive a phone call? The funniest one to me was in Murder in Malibu when the housekeeper comes in and she is dismissed by Jess, only to find out there is a phone call or the Lt.