It's an extremely knotty question — and so this review (open access) of the issue just published in New Phytologist is extremely welcome. Increased CO2 should be a fertilizer, and things grow quicker in warm weather, but as temperature increases, the rate of respiration rises more quickly than the rate of photosynthesis, so the rate at which trees release more carbon rises more quickly than the rate at which they soak it up (this also has implications for metabolic ecology: by understanding these effects, via metabolic rate, you can build a bridge between cellular and individual metabolism and the workings of the global carbon cycle). Also, plant growth is limited by other things, so a lack of nitrogen, for example, may limit trees' ability to respond to higher CO2, or warmer temperatures.
The review, which looks at studies in boreal and temperate forests, concludes that we don't really know what's going to happen. Here's what they say…
It is not in doubt that newly established young forests will continue to be C sinks for the foreseeable future. The key question is whether the mature forests that are C sinks today will continue to be sinks as the climate changes. … Forest ecosystem models indicate that the additional terrestrial sink arising from global climate change is likely to be maintained in the short term (over several decades), but may gradually diminish in the medium term. … Because of current limitations on our understanding with respect to acclimation of the physiological processes, the climatic constraints, and feedbacks among these processes – particularly those acting at the biome scale – projections of C-sink strengths beyond a few decades are highly uncertain.
This seems a good argument against tree-planting carbon offset schemes.