CO2 is a greenhouse gas. An increase in CO2 in the atmosphere from the pre-industrial level C0 = 280 ppmv to a concentration C will increase thermal radiation by dF W/m2 where  http://climatechangenationalforum.org/cause-and-effect-by-scott-denning, Scott Denning, "Cause and Effect", Climate Change National Forum, 2014-03-02
dF = 5.35 * ln(C/C0)
To see how such a thermal radiation change dF affects the global mean temperature, one can look at changes in the past, for example volcanic eruptions, and see how thermal radiation and temperature correlated. These examinations, as well as others, point to a climate sensitivity such that a thermal radiation increase of 1 W/m2 will increase the temperature by around 0.8 °C. [2,3,4,5]
A doubling of the CO2 concentration would mean a temperature increase of about 5.35*ln(560/280)*0.8 or about 3.0 °C from the pre-industrial level, which is often how climate sensitivity is presented. A tripling gives an increase of about 4.7 °C. Here are CO2 concentrations since 1780 and the resulting temperature change:
CDIAC and NOAA.
Following are atmospheric CO2 concentrations for the last 11,000 years:
NOAA, CDIAC and NOAA.
Projected CO2 concentrations and resulting temperature change are shown below. The CO2 projections come from the RCP 8.5 "business as usual" scenario, which, among the more well-known scenarios, has matched actual CO2 levels the closest so far, although the scenario could very well deviate later on of course, particularly on coal use perhaps.
NOAA and RCP 8.5.
The probability distribution function (PDF) for the climate sensitivity from the latest extensive IPCC report is shown below:
 http://www.skepticalscience.com/climate-sensitivity-advanced.htm, Dana Nuccitelli, "How sensitive is our climate?", Skeptical Science
 Reto Knutti and Gabriele C. Hegerl, 2008-10-26, "The equilibrium sensitivity of the Earth's temperature to radiation changes". Nature Geoscience 1 (11): 735–743, http://www.iac.ethz.ch/people/knuttir/papers/knutti08natgeo.pdf