To be ecologically truly sustainable, no live trees should be cut. That will most closely mimic a natural ecosystem instead of turning them into managed tree plantations (which ain't properly forests).
"Only take dead wood"
Now forests naturally create also a lot of biomass in the form of dead coarse wood, dead fine wood and dead leaves. That would slowly release its carbon as CO2 through natural decay, so we might as well collect and burn that. We can extract (let's say) half of that, while leaving the other half as habitat for animals etc.:
Dead wood, whether standing (snags) or down, plays an important role in forest ecology, acting as habitat for animals such as cavity-nesting birds, plants and microbial lives. (source: Carbon sequestration via wood burial (2008))
Also let's say we only would take the coarse wood (typically understood as having 10 cm stem diameter and up), as that is easiest to handle. In that case, a rough estimation can be based on this:
For example, a 1 km × 1 km area (100 hectares) would accumulate about 100 tonne of carbon per year for a typical coarse wood production rate of 0.1 kgC m-2 y-1 […]. 500 tonnes of carbon [are equivalent to] […] about 1000 tonne dry wood mass […]. (source: Carbon sequestration via wood burial (2008))
So 100 t y-1 km-2 of carbon yield from coarse deadwood in a mature forest, based on the naturally occurring tree deaths. Which is equivalent to about 200 t y-1 km-2 of dry wood mass.
Finally, taking out only half of that wood results in a production of 100 t y-1 km-2 of dry wood mass.
Comparing results
For comparability with the answer by m.w.jacobsen: my number of 100 t y-1 km-2 is equivalent to 1 t y-1 ha-1, while m.w.jacobsen had 5.4 t y-1 ha-1, both of dry wood mass.
The differences are that (1) I only include half the coarse deadwood (otherwise it's 2 t y-1 ha-1 when including all coarse deadwood, or roughly 4 t y-1 ha-1 when also including fine deadwood), (2) I don't include any clearcutting or cutting of live trees – but it is not too clear what impact that has on forest productivity, if any and (3) the numbers from the study I quote are quite rough. They are derived from global average forest productivity instead of that for only the temperate or tropic climate zone. Boreal forests have a lower productivity (see), lowering the average.