We present results from a high-resolution X-ray imaging study of nearby LINERs observed by ACIS on board Chandra. This study complements and extends previous X-ray studies of LINERs, focusing on the underexplored population of nearby dust-enshrouded infrared-bright LINERs. The sample consists of 15 IR-bright LINERs (LFIR/LB > 3), with distances that range from 11 to 26 Mpc. Combining our sample with previous Chandra studies, we find that ~51% (28/55) of the LINERs display compact hard X-ray cores. The nuclear 2-10 keV luminosities of the galaxies in this expanded sample range from ~2 × 1038 to ~2 × 1044 ergs s-1. We find that the most extreme IR-faint LINERs are exclusively active galactic nuclei (AGNs). The fraction of LINERs containing AGNs appears to decrease with IR brightness and increase again at the highest values of LFIR/LB. We find that of the 24 LINERs showing compact nuclear hard X-ray cores in the expanded sample that were observed at Hα wavelengths, only eight actually show evidence of a broad line. Similarly, of the 14 LINERs showing compact nuclear hard X-ray cores with corresponding radio observations, only eight display a compact flat spectrum radio core. These findings emphasize the need for high-resolution X-ray imaging observations in the study of IR-bright LINERs. Finally, we find an intriguing trend in the Eddington ratio versus LFIR and LFIR/LB for the AGN-LINERs in the expanded sample that extends over 7 orders of magnitude in L/LEdd. This correlation may imply a link between black hole growth, as measured by the Eddington ratio, and the star formation rate, as measured by the far-IR luminosity and IR-brightness ratio. If the far-IR luminosity is an indicator of the molecular gas content in our sample of LINERs, our results may further indicate that the mass accretion rate scales with the host galaxy's fuel supply. We discuss the potential implications of our results in the framework of black hole growth and AGN fueling in low-luminosity AGNs.