Traditionally stellar radiation has been the only heat source considered capable of determining global climate on long timescales. Here we show that terrestrial exoplanets orbiting low-mass stars may be tidally heated at high enough levels to induce a runaway greenhouse for a long enough duration for all the hydrogen to escape. Without hydrogen, the planet no longer has water and cannot support life. We call these planets "Tidal Venuses," and the phenomenon a "tidal greenhouse." Tidal effects also circularize the orbit, which decreases tidal heating. Hence, some planets may form with large eccentricity, with its accompanying large tidal heating, and lose their water, but eventually settle into nearly circular orbits (i.e. with negligible tidal heating) in the habitable zone (HZ). However, these planets are not habitable as past tidal heating desiccated them, and hence should not be ranked highly for detailed follow-up observations aimed at detecting biosignatures. Planets orbiting stars with masses <0.3 solar masses may be in danger of desiccation via tidal heating. We apply these concepts to Gl 667C c, a ~4.5 Earth-mass planet orbiting a 0.3 solar mass star at 0.12 AU. We find that it probably did not lose its water via tidal heating as orbital stability is unlikely for the high eccentricities required for the tidal greenhouse. As the inner edge of the HZ is defined by the onset of a runaway or moist greenhouse powered by radiation, our results represent a fundamental revision to the HZ for non-circular orbits. In the appendices we review a) the moist and runaway greenhouses, b) hydrogen escape, c) stellar mass-radius and mass-luminosity relations, d) terrestrial planet mass-radius relations, and e) linear tidal theories. [abridged]

59 pages, 11 figures, accepted to Astrobiology. New version includes an appendix on the water loss timescale