I've read not long ago that the absorption line of the $\text{CO}_2$ becomes larger with temperature thanks to Doppler effect (if it's false it would be great to tell me). My question is the following: Does this imply, like I think, that more energy is "stored" on Earth (thanks to greenhouse effect) and so play a role in climate change?
If the answer is yes, how big is this role?
No. Not in a significant way.
The full width at half maximum (FWHM) of a spectral line widened by thermal Doppler broadening is
$\Delta f_{\text{FWHM}}={\sqrt{\frac{8kT\ln(2)}{mc^{2}}}}f_{0}$
where $m$ is the mass of the emitting particle, $T$ is the temperature, $f_{0}$ is the frequency of the spectral line, and $k$ is the Boltzmann constant.
Let's look at the CO$_2$ absorption lines:
(Image from Wikipedia commons, source)
Let's take for example the band roughly between 12 and 20 $\mu m$. That corresponds to frequencies from 15 to 25 THz, with a mean of 20 THz.
If we solve the above equation for the mass of a CO$_2$ molecule ($44.01/6.02\times 10^{23}$ g), a frequency of 20 THz and an Earth's mean temperature of let's say 15°C (288.15 K) we get 1158 kHz broadening, and for an scenario 2°C warmer it would be 1162 kHz.
Therefore, the width of the absorption band would grow 8 kHz (4 kHz on either side). Which considering the original width of 10 THz corresponds to a 0.00008% increase. This could be roughly equated to an analogous increase in absorbed energy, which is absolutely negligible for all practical effects.
