Its yearly co2 consumption that they are responsible for sustaining. The plants are already consuming co2, this way that will continue to be the case.
At best, a young and growing forest can be a temporary carbon sink, until it reaches maturity (and steady state).
Alternatively, a forest might support continual carbon sinking if and only if the recently dead and/or mature trees were somehow harvested and treated/stored in such a way that their carbon would not decompose and re-enter the rest of the biosphere.
The only thing comparable these days is peat bogs that act as carbon sinks (because their plant matter does not decompose due to acidic and anaerobic environment).
A growing forest does consume CO2 more than it releases, but it does not seem to be the case now