The reason wood is considered neutral is, that most of the wood is used for construction and other stuff. If you burn it and happen to burn fossil fuel as well, nothing is neutral.
Over the life cycle of the tree from planting to burning, it's carbon neutral, though the shape of it for any given tree is 20 years of carbon sequestration followed by 20 minutes of carbon release.
1. Not nearly enough trees are being planted to offset the ones being cut down.
2. Neutral on a scale of 30 years doesn't much matter when we have to reduce the degree to which we're accelerating a highly non-linear climate change process right now.
Talking US specific, https://en.wikipedia.org/wiki/Deforestation_in_the_United_St... gives peaks and valleys depending on which time frame but a distinct decrease each year since 1963 with exception during the year of 1997. It is not flat.
Growing a tree, cutting it down, and burning the wood is carbon-neutral. But it's very dirty when it comes to other pollutants. A catalytic converter can help.
How is the wood cut? How is the wood transported? How is the wood processed?
Every one of these steps add carbon to the cycle. Some steps are minimal, others are huge; it all depends on scope.
I've heat my home with a wood-burning stove for 20-odd years. I have my own acreage and friends/family/farmers all contribute. Cutting down a tree, bringing it home, chopping it size so it fits in the stove are VERY energy intensive operations.
It isn’t though. A lot of energy is used planting then maintaining a forest for 30 years, cutting it down, processing it then shipping it. When timber is used in the building industry it’s a bit better but it is still notoriously wasteful. A lot of material goes to site and is binned - offcuts, wastage, over order and material in the wrong place.