Feedbacks that can destabilize the climates of synchronously-rotating rocky planets may arise on planets with strong day-night surface temperature contrasts. Earth-like habitable-zone (HZ) planets maintain stable surface liquid water over geological time. This requires equilibrium between the temperature-dependent rate of greenhouse-gas consumption by weathering,and greenhouse-gas resupply by other processes. Detected small-radius exoplanets, and anticipated M-dwarf HZ rocky planets, are expected to be tidally locked. We investigate two feedbacks that can destabilize climate on tidally-locked planets. (1) If small changes in pressure alter the temperature distribution across a planet's surface such that the weathering rate increases when the pressure decreases, a positive feedback occurs involving increasing weathering rate near the substellar point, decreasing pressure, and increasing substellar surface temperature. (2) When decreases in pressure increase the surface area above the melting point (through reduced advective cooling of the substellar point), and the corresponding increase in volume of liquid causes net dissolution of the atmosphere, a further decrease in pressure occurs. We use an idealized energy balance model to map out the conditions under which these instabilities may occur. The weathering runaway can shrink the habitable zone, and cause geologically rapid 10^3-fold pressure shifts within the HZ. Mars may have undergone a weathering runaway in the past. Substellar dissolution is usually a negative feedback or weak positive feedback on changes in pressure. Both instabilities are suppressed if the atmosphere has a high radiative efficiency. Our results are most relevant for atmospheres that are thin and have low greenhouse-gas radiative efficiency. These results identify a new pathway by which HZ planets can undergo rapid climate shifts and become uninhabitable. ; Comment: 30 pages, 6 figures, submitted to ApJ