Investigating the reliability of the assumption of instantaneous chemical equilibrium (ICE) for calculating the CO number density in the solar atmosphere is of crucial importance for the resolution of the long-standing controversy over the existence of ``cool clouds'' in the chromosphere and for determining whether the cool gas owes its existence to CO radiative cooling or to a hydrodynamical process. Here we report the first results of such an investigation in which we have carried out time-dependent gas-phase chemistry calculations in radiation hydrodynamical simulations of solar chromospheric dynamics. We show that while the ICE approximation turns out to be suitable for modeling the observed infrared CO lines at the solar disk center, it may substantially overestimate the ``heights of formation'' of strong CO lines synthesized close to the edge of the solar disk, especially concerning vigorous dynamic cases resulting from relatively strong photospheric disturbances. This happens because during the cool phases of the hydrodynamical simulations, the CO number density in the outer atmospheric regions is smaller than what is stipulated by the ICE approximation, resulting in decreased CO opacity in the solar chromosphere. As a result, the cool CO-bearing gas that produces the observed molecular lines must be located at atmospheric heights not greater than ~700 km. We conclude that taking into account the nonequilibrium chemistry improves the agreement with the available on-disk and off-limb observations but that the hydrodynamical simulation model has to be even cooler than anticipated by the ICE approximation, and this has to be the case at the ``new'' (i.e., deeper) formation regions of the rovibrational CO lines.