A unit of fuel must be in the fuel (bottom) slot for the furnace to burn the fuel already in it.  Example one.  One iron ore in top.  Two sticks in the bottom.  The furnace will burn the first stick and halfway smelt the ore, then start burning the second stick, but the smelting stops and will not finish.  Example two.  Eight cobble on top.  One coal in the bottom.  The coal will start burning, but the cobble will not be processed.  About half way through and added a log to the bottom.  The cobble started turning to stone, but one four were processed.  The furnace started burning the log, but the cobble no longer was getting processed.