We present comprehensive X-ray, optical, near- and mid-infrared and submm observations of GRB 080207 and its host galaxy. The afterglow was undetected in the optical and near-infrared (nIR) implying an X-ray-to-optical spectral slope less than 0.3, identifying GRB 080207 as a dark burst. Swift X-ray observations show extreme absorption in the host, which is confirmed by the unusually large optical extinction found by modelling the X-ray to nIR afterglow spectral energy distribution. Our Chandra observations obtained 8 d post-burst allow us to place the afterglow on the sky to subarcsec accuracy, enabling us to pinpoint an extremely red galaxy (ERO), with R-K > 5.4 (g-K˜ 7.5, VEGAmag) at the afterglow location. Follow-up host observations with the Hubble Space Telescope, Spitzer Space Telescope, Gemini, Keck and the James Clerk Maxwell Telescope provide a photometric redshift solution of ? (1σ, 1.56 <z <2.08 at 2σ) for the ERO host, and suggest that it is a massive and morphologically disturbed ultraluminous infrared galaxy system, with LFIR˜ 2.4 × 1012 L&sun;. These results add to the growing evidence that gamma-ray bursts (GRBs) originating in very red hosts always show some evidence of dust extinction in their afterglows (though the converse is not true - some extinguished afterglows are found in blue hosts). This indicates that a poorly constrained fraction of GRBs occurs in very dusty environments. By comparing the inferred stellar masses, and estimates of the gas phase metallicity in both GRB hosts and submm galaxies we suggest that many GRB hosts, even at z > 2, are at lower metallicity than the submm galaxy population, offering a likely explanation for the dearth of submm-detected GRB hosts. However, we also show that the dark GRB hosts are systematically more massive than those hosting optically bright events, perhaps implying that previous host samples are severely biased by the exclusion of dark events.