Supernova (SN) 2006gy was a hydrogen-rich core-collapse SN that remains one of the most luminous optical SNe ever observed. The total energy budget (>2 × 10$^51$ erg radiated in the optical alone) poses many challenges for standard SN theory. We present new ground-based near-infrared (NIR) observations of SN 2006gy, as well as a single epoch of Hubble Space Telescope (HST) imaging obtained more than two years after the explosion. Our NIR data taken around peak optical emission show an evolution that is largely consistent with a cooling blackbody, with tentative evidence for a growing NIR excess starting around day åisebox-0.5ex 130. Our late-time Keck adaptive optics NIR image, taken on day 723, shows little change from previous NIR observations taken around day 400. Furthermore, the optical HST observations show a reduced decline rate after day 400, and the SN is bluer on day 810 than it was at peak. This late-time decline is inconsistent with $^56$Co decay, and thus is problematic for the various pair-instability SN models used to explain the nature of SN 2006gy. The slow decline of the NIR emission can be explained with a light echo, and we confirm that the late-time NIR excess is the result of a massive (gsim10 M $_sun$) dusty shell heated by the SN peak luminosity. The late-time optical observations require the existence of a scattered light echo, which may be generated by the same dust that contributes to the NIR echo. Both the NIR and optical echoes originate in the proximity of the progenitor, i̊sebox-0.5ex 10$^18$ cm for the NIR echo and lsim10-40 pc for the optical echo, which provides further evidence that the progenitor of SN 2006gy was a very massive star.