The High Altitude Water Cherenkov (HAWC) observatory is a wide field-of-view detector sensitive to gamma rays of 100 GeV to a few hundred TeV. Located in central Mexico at 19° North latitude and 4100 m above sea level, HAWC will observe gamma rays and cosmic rays with an array of water Cherenkov detectors. The full HAWC array is scheduled to be operational in Spring 2015. In this paper, we study the HAWC sensitivity to the gamma-ray signatures of high-mass (multi-TeV) dark matter annihilation. The HAWC observatory will be sensitive to diverse searches for dark matter annihilation, including annihilation from extended dark matter sources, the diffuse gamma-ray emission from dark matter annihilation, and gamma-ray emission from nonluminous dark matter subhalos. Here we consider the HAWC sensitivity to a subset of these sources, including dwarf galaxies, the M31 galaxy, the Virgo cluster, and the Galactic center. We simulate the HAWC response to gamma rays from these sources in several well-motivated dark matter annihilation channels. If no gamma-ray excess is observed, we show the limits HAWC can place on the dark matter cross section from these sources. In particular, in the case of dark matter annihilation into gauge bosons, HAWC will be able to detect a narrow range of dark matter masses to cross sections below thermal. HAWC should also be sensitive to nonthermal cross sections for masses up to nearly 1000 TeV. The constraints placed by HAWC on the dark matter cross section from known sources should be competitive with current limits in the mass range where HAWC has similar sensitivity. HAWC can additionally explore higher dark matter masses than are currently constrained.