The uncertainty surrounding the nature of the heating mechanism for the dust that emits at mid- to far-IR (MFIR) wavelengths in active galaxies limits our understanding of the links between active galactic nuclei (AGNs) and galaxy evolution, as well as our ability to interpret the prodigious infrared and submillimeter emission of some of the most distant galaxies in the universe. Here we report deep Spitzer observations of a complete sample of powerful, intermediate-redshift (0.05 <z <0.7) radio galaxies and quasars. We show that AGN power, as traced by [O III] λ5007 emission, is strongly correlated with both the mid-IR (24 μm) and the far-IR (70 μm) luminosities, but with increased scatter in the 70 μm correlation. A major cause of this increased scatter is a group of objects that falls above the main correlation and shows signs of prodigious recent star formation activity at optical wavelengths, along with relatively cool MFIR colors. These results provide evidence that illumination by the AGNs is the primary heating mechanism for the dust emitting at both 24 and 70 μm, with starbursts dominating the heating of the cool dust in only 20%-30% of objects. This implies that powerful AGNs are not always accompanied by the type of luminous starbursts that are characteristic of the peak of activity in major gas-rich mergers.