We present an analysis of deep mid- to far-infrared (MFIR) Spitzer photometric observations of the southern 2Jy sample of powerful radio sources (0.05 <z <0.7), conducting a statistical investigation of the links between radio jet, active galactic nucleus (AGN), starburst activity and MFIR properties. This is part of an ongoing extensive study of powerful radio galaxies that benefits from both complete optical emission line information and a uniquely high detection rate in the far-infrared (far-IR). We find tight correlations between the MFIR and [O III]λ5007 emission luminosities, which are significantly better than those between MFIR and extended radio luminosities, or between radio and [O III] luminosities. Since [O III] is a known indicator of intrinsic AGN power, these correlations confirm AGN illumination of the circumnuclear dust as the primary heating mechanism for the dust producing thermal MFIR emission at both 24 and 70 μm. We demonstrate that AGN heating is energetically feasible, and identify the narrow-line region clouds as the most likely location of the cool, far-IR emitting dust. Starbursts make a major contribution to the heating of the cool dust in only 15%-28% of our targets. We also investigate the orientation dependence of the continuum properties, finding that the broad- and narrow-line objects in our sample with strong emission lines have similar distributions of MFIR luminosities and colors. Therefore our results are entirely consistent with the orientation-based unified schemes for powerful radio galaxies. However, the weak line radio galaxies form a separate class of objects with intrinsically low-luminosity AGNs in which both the optical emission lines and the MFIR continuum are weak.