I have an auto smelter setup to convert my cobblestone to stone, when I tried to smelt the stone obtained from the auto smelter no action would start. When I went to other furnaces that had cobblestone smelted into stone, those stone blocks could be run through the furnaces again to be converted to smooth stone. 

As an addition to this, it appears that if the stone was left in the furnace, it could be smelted again to smooth stone, but any stone that was in a chest or inventory cannot