Most carbonaceous fuels leave behind a small amount of ashes after full combustion. The amount depends on the grade of the fuel (lignite vs anthracite for example). Current sources of ashes in Pyrotech are currently the campfire, pit burning, pit kiln and the bloomery as far as I'm aware.

The brick and refractory machines do consume solid fuels, but don't generate ashes. Thus the suggestion to have ashes generate in the firebox of these machines that needs to be cleared out, similar to the campfire.