A Dumbwaiter is a small, square elevator used for raising and lowering objects (and sometimes people). There are several dumbwaiters in Problem Sleuth.
Two dumbwaiters, stacked one on top of the other, allow Problem Sleuth and Ace Dick swap small items back and forth without actually visiting the other's office. The dumbwaiter shaft ascends all the way up to just outside Mobster Kingpin's office, covered by the dreaded Bowen Stilson Dogg. This shaft, the bust, and presumably both dumbwaiters were very thoroughly demolished in a Hot Sauce-related explosion.
A third dumbwaiter, made of peanut brittle, connects the Imaginary City Streets with the Rooftop. Pickle Inspector summoned this object to make navigation through the imaginary universe a little easier.