a very clever question shsnawada!
in simple physics ... get some wood .. dip that in a beaker full to brim with water ... measure the volume of displaced water .. that would be volume of the wood .. now let it dry and ...put it on scales .. take exact reading! ..
put a glass dome over the wood covering it .. leave a whole for oxygen at the bottom of the sphere .. and yes to determine the volume of the glass dome ..(4/6 pi.r^3)
.. now light the wood .. note down the EXACT reading on teh scale at a particular instant and quickly kill the fire .. now weight theleft over wood with ashes ...
density of fire (approx) should be(in Kg/m^3) = (((mass of wood (unburnt) - (weight of wood +ashes) - reading on scale while wood was burning))/9.8)/ (volume of dome - volume of wood)
i think it should work .. but then again .. i could have missed something ..