Rest mass is not conserved, the total energy is. Rest mass is just one of
the ways in which the energy of a system can be expressed. For example,
when an electron and positron annihilate, their mass-energy appears as
the energy of the emitted, rest-massless photons.
An example analogous to burning is the fusion of 4 hydrogens to form
helium. A look at the periodic chart will show that the mass of He4 is
less than that of 4 H atoms. The difference is the large, negative,
binding energy of baryons in He4, which is given off in the process. The
actual fusion is a multi-step process, the details of which I don't
recall, but the excess energy is carried off by photons and neutrinos.
Ordinary burning is similar, but the binding energy is electro-magnetic
and much less, so that the reduction in rest mass is so small as to be,
I'm sure, undetectable.
Al