We present new theoretical period-luminosity-metallicity (PLZ) relations for RR Lyrae stars (RRLs) at Spitzer and WISE wavelengths. The PLZ relations were derived using nonlinear, time-dependent convective hydrodynamical models for a broad range of metal abundances (Z=0.0001-0.0198). In deriving the light curves, we tested two sets of atmospheric models and found no significant difference between the resulting mean magnitudes. We also compare our theoretical relations to empirical relations derived from RRLs in both the field and in the globular cluster M4. Our theoretical PLZ relations were combined with multi-wavelength observations to simultaneously fit the distance modulus, {mu}0, and extinction, A_V_, of both the individual Galactic RRL and of the cluster M4. The results for the Galactic RRL are consistent with trigonometric parallax measurements from Gaia's first data release. For M4, we find a distance modulus of {mu}0=11.257+/-0.035mag with A_V_=1.45+/-0.12mag, which is consistent with measurements from other distance indicators. This analysis has shown that, when considering a sample covering a range of iron abundances, the metallicity spread introduces a dispersion in the PL relation on the order of 0.13mag. However, if this metallicity component is accounted for in a PLZ relation, the dispersion is reduced to ~0.02mag at mid-infrared wavelengths.