We present observations and analysis of the host galaxies of 23 heavily dust-obscured gamma-ray bursts (GRBs) observed by the Swift satellite during the years 2005-2009, representing all GRBs with an unambiguous host-frame extinction of A$_V$ > 1 mag from this period. Deep observations with Keck, Gemini, Very Large Telescope, Hubble Space Telescope, and Spitzer successfully detect the host galaxies and establish spectroscopic or photometric redshifts for all 23 events, enabling us to provide measurements of the intrinsic host star formation rates, stellar masses, and mean extinctions. Compared to the hosts of unobscured GRBs at similar redshifts, we find that the hosts of dust-obscured GRBs are (on average) more massive by about an order of magnitude and also more rapidly star forming and dust obscured. While this demonstrates that GRBs populate all types of star-forming galaxies, including the most massive, luminous systems at z ≈ 2, at redshifts below 1.5 the overall GRB population continues to show a highly significant aversion to massive galaxies and a preference for low-mass systems relative to what would be expected given a purely star-formation-rate-selected galaxy sample. This supports the notion that the GRB rate is strongly dependent on metallicity, and may suggest that the most massive galaxies in the universe underwent a transition in their chemical properties åisebox-0.5ex 9 Gyr ago. We also conclude that, based on the absence of unobscured GRBs in massive galaxies and the absence of obscured GRBs in low-mass galaxies, the dust distributions of the lowest-mass and the highest-mass galaxies are relatively homogeneous, while intermediate-mass galaxies (i̊sebox-0.5ex 10$^9$ M $_☉$) have diverse internal properties.