We report 18 cm VLBI continuum imaging observations at ~5 mas resolution for UGC 5101, NGC 7469, and Mrk 231, all part of a sample of luminous infrared galaxies that were shown to have strong VLBI radio cores in a previous 18 cm VLBI survey. Mrk 231, generally considered to be a dust-enshrouded quasi-stellar object (QSO), shares many characteristics with quasars, including a broad Sy1 optical emission spectrum, optical and radio variability, and a broad absorption line spectrum. NGC 7469 is a classical Sy1 galaxy, and UGC 5101 is a LINER system. The radio morphology of these three systems on VLBI scales is like that of an active galactic nucleus (AGN), with well-defined ridgelines and high-brightness yet spatially resolved components. The structure and flux densities of these VLBI components are not consistent with starburst-generated radio supernovae of the type found in Arp 220. On scales of ~100 pc and perhaps beyond, the radio continuum in all three objects appears to be dominated by an AGN, not a starburst. Radio emission on larger scales may well originate, in part or in total, in a less compact circumnuclear star-forming region. This is in contrast to the situation thought to prevail in many luminous infrared galaxies (LIGs), which involves a very compact and dense nuclear star-forming region. By placing a lower limit on the AGN-related radio emission, it is possible to shed light on the relative luminosities of the AGN and starburst in these objects. Using the far-infrared-radio correlation for starburst-related radio emission, we show that, despite the unambiguous evidence for AGN activity in these systems, the AGN luminosity, although energetically important, may be exceeded by the extended starburst luminosity in all three cases, unless the bolometric luminosity of the entire system has been underestimated. The lower limits on AGN-related radio emission indicate that these AGNs are somewhat overluminous in the radio compared with radio-quiet Palomar-Green (PG) QSOs and thus that the AGN radio luminosity cannot be used to estimate the bolometric luminosity of the AGN. However, comparing near- and mid-IR luminosities of the LIGs to those of PG QSOs, we find evidence that a significant fraction of the bolometric luminosity of the AGN in Mrk 231 and to a lesser extent UGC 5101 may be radiated away from Earth and thus not detected at any wavelength. The VLBI structure observed in Mrk 231 allows additional interpretation. Confirming and extending VLBI imaging by Ulvestad, Wrobel, and Carilli, our continuum image shows a triple structure, with a core and two lobes, causing it to be classied as a compact symmetric object. It has been suggested that these sources are young, τ 10⁶ yr, with the hot spots representing the working surface of a relativistic jet on the ambient medium. If the southern (primary) lobe/hot-spot in Mrk 231 is confined by ram pressure, we estimate a lobe advance speed, v_a ~ 10^-4c, and an age for the jet/compact source, τ < 10⁶ yr. We have also imaged the 1667 MHz OH maser emission in Mrk 231, which is extended on scales of 50-100 mas (40-80 pc) and probably coincides with the inner region of the disk, which is seen in CO emission and H I absorption. Among OH megamaser sources studied at high sensitivity with milliarcsecond resolution, Mrk 231 is unique in the stringent upper limits placed on the flux density of compact OH structures of the type found in Arp 220 and other LIGs. It is possible that the circumnuclear environment of Mrk 231 has been sufficiently disrupted by the emergent QSO that the cool dense clouds necessary for such compact masers no longer exist.