Everything wooden is supposed to work as fuel in a furnace, in the bowl's case it does not. I tested on versions 1.6.4 and up and it does not work.
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CubeTheThird
Duplicate of MC-9691 - If you have not, please use the search function in the future, to see if your bug has already been submitted. If you could not find the original report, please comment with the keywords you searched for.
There are several other wooden things that cannot burn: "Boats, Doors, Wooden buttons, Signs, Ladders, perhaps Bows and Fishing rods"