Biosphere carbon is carbon neutral. If you grow a tree, pulp the tree, and burn the tree, the amount of overall carbon available has not changed as the process of growing new trees will reuse the same carbon. Ignoring external energy costs, obvs.
It's a little weird to think about but those wood burning plants are mostly taking the production waste of lumber processing and using it for fuel, similar to wood pellet grills but much larger, and it burns surprisingly clean, relatively speaking anyway.
What's killing us is pulling billions of tons of carbon that was buried deep in the earth and reinjecting that into the atmosphere.