Using wood to burn for heating IS carbon neutral. ALL of the carbon in a tree comes from CO2 from the air. ALL of the carbon in that wood IS converted by complete combustion to CO2 into the air again. The SLIGHT exception if you do have a good stove / furnace that burns properly is that a SMALL amount of that carbon may end up as other combustion products (gaseous or solid) that do degrade over a longer time period to carbon / oxygen gases OR to other nutrients in the soil that are likely to end up back in plants.
Like any other forest resource, if you plan to use a woodlot full of trees as a fuel source, it needs to be managed on a Perpetual Yield basis. That is, over the long term, the rate of harvesting for whatever uses must match the rate of re-growth. That re-growth rate depends a lot on the climate in your area and the species of trees. I live in the Pre-cambrian Shield area of Canada where many natural forests are dominated by coniferous Black and White Spruces, Jackpine, some Balsam Fir, mixed with several deciduous species like Birch and White and Black Poplar (Aspen). Most of the coniferous species are considered mature and suitable for harvest for many uses at 70 to 90 years, and over-mature beyond that. The deciduous species are mature in 40 to 75 years. So VERY roughly, whatever your planned rate of harvesting is per year, you need a woodlot of about 70 t0 90 times that to be able to manage on a perpetual yield basis.
We are NOT used to thinking in those terms for forests and woodlots, because the time frame is very much like our own lifetime. BUT the principle is exactly the same as a farm field, where the yearly harvest is exactly the same as the yearly growth. In that latter case we don't worry about the time frames - that field may generate harvests every year for our entire lives.
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