  The enormous increase in mid-IR sensitivity and spatial and spectral
resolution provided by the JWST spectrographs enables, for the first time,
detailed extragalactic studies of molecular vibrational bands. This opens an
entirely new window for the study of the molecular interstellar medium in
luminous infrared galaxies (LIRGs). We present a detailed analysis of
rovibrational bands of gas-phase CO, H$_2$O, C$_2$H$_2$ and HCN towards the
heavily-obscured eastern nucleus of the LIRG VV 114, as observed by NIRSpec and
MIRI MRS. Spectra extracted from apertures of 130 pc in radius show a clear
dichotomy between the obscured AGN and two intense starburst regions. We detect
the 2.3 $\mu$m CO bandheads, characteristic of cool stellar atmospheres, in the
star-forming regions, but not towards the AGN. Surprisingly, at 4.7
$\mathrm{\mu}$m we find highly-excited CO ($T_\mathrm{ex} \approx 700-800$ K
out to at least rotational level $J = 27$) towards the star-forming regions,
but only cooler gas ($T_\mathrm{ex} \approx 200$ K) towards the AGN. We
conclude that only mid-infrared pumping through the rovibrational lines can
account for the equilibrium conditions found for CO and H$_2$O in the
deeply-embedded starbursts. Here the CO bands probe regions with an intense
local radiation field inside dusty young massive star clusters or near the most
massive young stars. The lack of high-excitation molecular gas towards the AGN
is attributed to geometric dilution of the intense radiation from the bright
point source. An overview of the relevant excitation and radiative transfer
physics is provided in an appendix.
