There are several key ways in which rising atmospheric CO₂ concentrations will affect photosynthesis, and these are related to the different types of photosynthesis. In order to properly answer your question, I'll provide some background about photosynthesis itself.
Photosynthesis evolved in a high-CO₂ atmosphere, before the oxygen-enrichment of the atmosphere (which actually happened as a result of photosynthesis). Most plant species operate C3 photosynthesis. In these plants, carbon dioxide diffuses into the cell where it is fixed by Ribulose-1,5-bisphosphate carboxylase oxygenase (RuBisCO) into a 3-carbon molecule (hence C3), which is then polymerised to make sugars. A crucial fact about RuBisCO is that it has both carboxylase (carbon-fixing) activity and oxygenase (oxygen-fixing) activity. This means that oxygen and carbon dioxide compete for the active site on the enzyme complex, leading to RuBisCO being quite inefficient and slow at fixing carbon in higher oxygen concentrations. That didn't matter in the high-CO₂ atmosphere of the early Earth, but in todays atmosphere O₂ concentrations are high enough that they severely limit the productivity of C3 plants.
However, plants haven't just been growing slowly all that time - several mechanisms for increasing photosynthetic efficiency have evolved. The most influential systems involve concentrating carbon dioxide in a particular area, excluding oxygen, and concentrating RuBisCO in that same area. This avoids the oxygen competition for the active site and allows RuBisCO to operate more efficiently. The key adaptation here is C4 photosynthesis - the system which is present in most grasses and many of the most productive plants on Earth (e.g. maize, sugarcane, Miscanthus). It has evolved at least 62 times independently. It works by having RuBisCO concentrated within 'bundle sheath' cells which are surrounded by a layer of suberin wax. This layer prevents CO₂ escaping and O₂ from getting in. CO₂ from the atmosphere is then fixed in different cells - 'mesophyll cells' - by another enzyme - Phosphoenolpyruvate carboxylase (PEPC), resulting in a four-carbon molecule (hence C4). This 4-carbon acid, (malate or oxaloacetate depending on the system) is then shuttled into the bundle sheath cells. There, the CO₂ is released again by a variety of enzymes depending on the system, creating a high CO₂ concentration in the cell where RuBisCO can then work efficiently.
In general, C4 plants are much (about 50%) more efficient than their C3 counterparts, and they are particularly well adapted to high temperatures and moist environments. So, to answer your first question: as atmospheric CO2 levels continue to rise, C3 plants will gradually be able to photosynthesise more efficiently. Interestingly though, C4 plants are predicted to also benefit from increased atmospheric CO₂. If global temperatures rise as predicted, both C3 and C4 plants will be able to operate more efficiently than they currently do, up to a maximum temperature beyond which enzymes will begin to denature faster and efficiency will drop. One consideration is that the difference in efficiency between C3 and C4 systems will decrease, which may significantly alter the makeup of plant communities around the world.
This is a vast oversimplification, but it is accurate for the predicted overall effects. Localised effects (i.e. productivity changes in a particular region or for a particular crop) will depend on habitat, physiology, etc.
Some key papers to launch you into the literature:
- Leakey, A.D.B., Bernacchi, C.J., Dohleman, F.G., Ort, D.R. & Long, S.P. (2004) Will photosynthesis of maize (Zea mays) in the US Corn Belt increase in future [CO2] rich atmospheres? An analysis of diurnal courses of CO2 uptake under free‐air concentration enrichment (FACE). Global Change Biology. [Online] 10 (6), 951–962. Available from: doi:10.1111/j.1529-8817.2003.00767.x [Accessed: 31 January 2012].
- Morgan, J.A., LeCain, D.R., Pendall, E., Blumenthal, D.M., Kimball, B.A., Carrillo, Y., Williams, D.G., Heisler-White, J., Dijkstra, F.A. & West, M. (2011) C4 grasses prosper as carbon dioxide eliminates desiccation in warmed semi-arid grassland. Nature. [Online] 476 (7359), 202–205. Available from: doi:10.1038/nature10274 [Accessed: 31 January 2012].