Carbon dioxide (CO²) is taken from the atmosphere and used to grow by trees. When they die and decay or are buried, this CO² is released back into the atmosphere. In a mature, unmanaged forest the amount of carbon dioxide being absorbed by growing trees is the same amount as what is being given off by decaying dead trees, animals, microbes etc, that live off the trees as they live and die.

Wood is never removed faster than it is added by new growth. Therefore the CO² released when the wood fuel is burned, is never anymore than the CO² being taken up by new growth.