We present JWST/MIRI spectra from the Medium-resolution Spectrometer of I Zw 18, a nearby dwarf galaxy with a metallicity of ∼3% Solar. Here, we investigate warm molecular hydrogen, H2, observed in spectra extracted in ∼120 pc apertures centered on eleven regions of interest. We detect seven H2 rotational lines, some of which are among the weakest ever measured. The H2 population diagrams are fit with local-thermodynamic-equilibrium models and models of photodissociation regions. We also fit the ortho-/para-H2 ratios (OPRs); in three of the six regions for which it was possible to fit the OPR, we find values significantly greater than 3, the maximum value for local thermodynamic equilibrium. To our knowledge, although predicted theoretically, this is the first time that OPR significantly >3 has been measured in interstellar gas. We find that an OPR tends to increase with decreasing H2 column density, consistent with the expected effects of self-shielding in advancing photodissociation fronts. The population diagrams are consistent with H nucleon densities of ∼105 cm−3, and an interstellar radiation field scaling factor, G0, of ∼103. This warm, dense H2 gas coexists with the same highly ionized gas that emits [O IV] and [Ne V]. Emission from T ≳ 50 K dust is detected, including an as-yet-unidentified dust emission feature near 14 μm; possible identification of Al2O3 is discussed. The continuum emission from several regions requires that a considerable fraction of the refractory elements be incorporated in dust. Despite stacking spectra in the SE where H2 is found, no significant emission from polycyclic aromatic hydrocarbons is detected.