The X-ray persistence and characteristically soft spectrum of the black hole X-ray binary LMC X-3 make this source a touchstone for penetrating studies of accretion physics. We analyze a rich, 10-year collection of optical/infrared (OIR) time-series data in conjunction with all available contemporaneous X-ray data collected by the ASM and PCA detectors aboard the Rossi X-ray Timing Explorer. A cross-correlation analysis reveals an X-ray lag of ~2 weeks. Motivated by this result, we develop a model that reproduces the complex OIR light curves of LMC X-3. The model is comprised of three components of emission: stellar light; accretion luminosity from the outer disk inferred from the time-lagged X-ray emission; and light from the X-ray-heated star and outer disk. Using the model, we filter a strong noise component out of the ellipsoidal light curves and derive an improved orbital period for the system. Concerning accretion physics, we find that the local viscous timescale in the disk increases with the local mass accretion rate; this in turn implies that the viscosity parameter alpha decreases with increasing luminosity. Finally, we find that X-ray heating is a strong function of X-ray luminosity below ~50% of the Eddington limit, while above this limit X-ray heating is heavily suppressed. We ascribe this behavior to the strong dependence of the flaring in the disk upon X-ray luminosity, concluding that for luminosities above ~50% of Eddington, the star lies fully in the shadow of the disk.