We report measurements of the thermal emission of the young and massive planet CoRoT-2b at 4.5 and 8 μm with the Spitzer Infrared Array Camera (IRAC). Our measured occultation depths are 0.510±0.042% at 4.5 and 0.41±0.11% at 8 μm. In addition to the CoRoT optical measurements, these planet/star flux ratios indicate a poor heat distribution on the night side of the planet and agree better with an atmosphere free of temperature inversion layer. Still, such an inversion is not definitely ruled out by the observations and a larger wavelength coverage is required to remove the current ambiguity. Our global analysis of CoRoT, Spitzer, and ground-based data confirms the high mass and large size of the planet with slightly revised values (M_p = 3.47±0.22 MJ, R_p = 1.466±0.044 RJ). We find a small but significant offset in the timing of the occultation when compared to a purely circular orbital solution, leading to e cosω = -0.00291±0.00063 where e is the orbital eccentricity and ω is the argument of periastron. Constraining the age of the system to at most a few hundred Myr and assuming that the non-zero orbital eccentricity does not come from a third undetected body, we modeled the coupled orbital-tidal evolution of the system with various tidal Q values, core sizes, and initial orbital parameters. For Q_s' = 105-106, our modeling is able to explain the large radius of CoRoT-2b if Q_p'≤ 105.5 through a transient tidal circularization and corresponding planet tidal heating event. Under this model, the planet will reach its Roche limit within 20 Myr at most. Based on data collected with the VLT/FORS2 instrument at ESO Paranal Observatory, Chile (programs 081.C-0413(B)). The photometric timeseries used in this work are only available in electronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/511/A3