Chemical burning is the ordinary kind of burning associated with the burning
of fossil fuels. In this case, the energy release occurs due to
the formation of molecules. For example, energy is released when
The snapping together of the particles leads to the energy release (similar to the case of gravitational energy release, except that now the energy comes from the elctrical force between the atoms). Note that this means that the energy from chemical burning has as its origin the electrical force.
To answer the question of whether the Sun could shine by chemical burning, we asume that the Sun is made purely out of C and O (a very generous assumption). Now, since C has 12 neutrons and protons and O has 16 neutrons and protons (in their most abundant isotopic forms), there are
N ~ M(sun)/(28 x mass of the proton) ~ 4 x 10**55 particles
of C and O in our hypothetical Sun. The total chemical energy is therefore 7 x 10**44 ergs. A large number to be sure but still over 1 trillion times too small to explain the energy output of the Sun!