A sample of 178 extragalactic objects is defined by correlating the 60 μm IRAS Faint Source Catalog (FSC) with the 5 GHz Parkes-MIT-NRAO (PMN) catalog. Of these, 98 objects lie above the radio/far-infrared relation for radio-quiet objects. These radio-excess galaxies and quasars have a uniform distribution of radio excesses, with no evidence for a radio-loud correlation equivalent to the well-known radio-quiet radio/far-infrared relation. The radio-intermediate objects appear to be a new population of active galaxies not present in previous radio/far-infrared samples chosen using more stringent far-infrared criteria. The radio-excess objects extend over the full range of far-infrared luminosities seen in extragalactic objects, from low-luminosity galaxies with νL_ν(60 μm) < 10⁹ L to ultraluminous infrared galaxies with νL_ν(60 μm) > 10¹² L. Objects with small radio excesses are more likely to have far-infrared colors similar to starbursts, while objects with large radio excesses have far-infrared colors typical of pure AGNs. Some of the most far-infrared luminous radio-excess objects have the highest far-infrared optical depths. These are good candidates to search for hidden broad-line regions in polarized light or via near-infrared spectroscopy. Some low far-infrared luminosity radio-excess objects appear to derive a dominant fraction of their far-infrared emission from star formation, despite the dominance of the AGN at radio wavelengths. Many of the radio-excess objects have sizes likely to be smaller than the optical host but show optically thin radio emission, rather than flat radio spectra indicative of compact quasar cores. We draw parallels between these objects and high radio luminosity compact steep-spectrum and gigahertz peaked-spectrum objects. Radio sources with these characteristics may be young AGNs in which the radio activity has begun only recently. Alternatively, high central densities in the host galaxies may be confining the radio sources to compact sizes. We discuss future observations required to distinguish between these possibilities and determine the nature of radio-excess objects.