I've heard several contradictory views and I'm not sure which factor do climatologists count with. Counting with all the effects of the current climate change (including changes of biotopes on places such as Siberia, as well as the higher percent of CO2 in the atmosphere), will the conditions help make better conditions for growing trees (I know the effect on biosphere is usually claimed to be synergic, e.i. for example coral reefs will have less CaCO3 to work with so they will make the global warming appear faster. So please consider just the average effect on trees)
I don't think anyone knows for sure. It will certainly depend on which spatial location, and which tree species, you're thinking of: while "global average effect on growth (e.g. measured by change in net ecosystem exchange of forested ecosystems) of all tree species" is at least well-defined, it will be an average across an enormously variable set of communities, and species within communities. Some of the relevant factors:
- if you count the direct effects of increased CO$_2$, this will generally increase growth rates, although it will be highly variable across ontogeny (life stage, i.e. seedling vs sapling vs adult) and species. In general changes in growth rate will be higher for seedlings and for early-successional/fast-growing species, which may in turn change community structure: see Bolker et al. 1995 for one synthetic study of this.
- increased temperature would be expected on average to increase growth rates
- decreased moisture availability (potential evapotranspiration) would be expected on average to decrease growth rates
- increased microbial activity (e.g., due to increased temperature and/or moisture availability) would be expected to lead on average to increased nutrient availability, increasing growth rates
- if you're just measuring "net biomass accrual to trees", then changes in large-scale disturbance (wildfire, windstorms, pest and pathogen outbreaks ...) will tend to decrease standing biomass (although it may also lead to temporary pulses of increased carbon accrual as forests re-grow after disturbance)
- note that almost all of these driving factors (temperature, precipitation, water availability, frequency of disturbance) will themselves vary a lot in space.
For a summary, you could start by looking at Purves and Pacala 2008 for some of the ways in which people are trying to put these pieces together.