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, H2O, C2H2, and HCN toward the heavily obscured eastern nucleus of the LIRG VV 114, as observed by NIRSpec and the medium resolution spectrograph on the Mid-InfraRed Instrument (MIRI MRS). Spectra extracted from apertures of 130 pc in radius show a clear dichotomy between the obscured active galactic nucleus (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 toward the AGN. Surprisingly, at 4.7 mu m, we find highly excited CO (T ex approximate to 700-800 K out to at least rotational level J = 27) toward the star-forming regions, but only cooler gas (T ex approximate to 200 K) toward the AGN. We conclude that only mid-infrared pumping through the rovibrational lines can account for the equilibrium conditions found for CO and H2O 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 toward 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.