Does the extension of the absorption line of $\text{CO}_2$ with temperature play a role in the climate change?
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.