GRB 051022 was undetected to deep limits in early optical observations, but precise astrometry from radio and X-rays showed that it most likely originated in a galaxy at z~0.8. We report radio, optical, near-infrared, and X-ray observations of GRB 051022. Using the available X-ray and radio data, we model the afterglow and calculate its energetics, finding it to be an order of magnitude lower than that of the prompt emission. The broadband modeling also allows us to precisely define various other physical parameters and the minimum required amount of extinction to explain the absence of an optical afterglow. Our observations suggest a high extinction, at least 2.3 mag in the infrared (J) and at least 5.4 mag in the optical (U) in the host-galaxy rest frame. Such high extinctions are unusual for GRBs and likely indicate a geometry where our line of sight to the burst passes through a dusty region in the host that is not directly colocated with the burst itself.