How does CO₂ concentration affect photosynthesis?
The rate-limiting step of photosynthesis is the CO2 assimilating enzyme Rubisco (short for ribulose-1,5-bisphosphate carboxylase/oxygenase) (Jensen, 2000). It uses ribulose-1,5-bisphosphate and CO2 as substrates to generate glucose. 

Given that Rubisco is the rate limiting step in photosynthesis, an increase in its substrate CO2 would expectedly lead to an increase in photosynthesis. However, the regulation of Rubisco is complex and is influenced not only by CO2, but also by O2 (which competes with CO2 for the active site), Mg2+ and a regulating enzyme called Rubisco activase (Jensen, 2000). Hence, the effects of an increase in atmospheric CO2 may be more complex than simply enhancing photosynthesis by increasing Rubisco activity.

Indeed, a review by Poorter (1993) showed that a doubling of CO2 lead to an average increase in photosynthesis of only 37% across more than 150 plant species. He describes various factors that determine photosynthesis rates other than CO2:  


One factor that limited photosynthesis under high CO2 was nitrogen (N). Since Rubisco is an enzyme, it has to be synthesized from amino acids. As Rubisco constitutes about 30% of the total protein in a plant leaf, Rubisco is probably the most abundant protein on earth and a major sink for plant nitrogen (Jensen, 2000). For example, C3 species capable of symbiosis with N2-fixing organisms had higher growth increases compared to other C3 species under high CO2; 
Certain plants utilize CO2 better than others. CAM species were the least responsive, followed by C4 plants, while C3 plants increased their photosynthesis rates the most under high CO2. Examples of C3 plants are herbaceous crop plants (Poorter, 1993);
Plants with a large intrinsic growth rate may benefit more from high CO2. These plants are said to have a high sink strength, as they rapidly convert additional photosynthesis products into outgrowth. 


References
- Jensen, PNAS (2000); 97(24): 12937–38
- Poorter, Veg (1993);104/105: 77-97