``Hard" massive black hole (MBH) binaries embedded in steep stellar cusps can shrink via three-body slingshot interactions. We show that this process will inevitably be accompanied by a burst of stellar tidal disruptions, at a rate that can be several orders of magnitude larger than that appropriate for a single MBH. Our numerical scattering experiments reveal that: 1) a significant fraction of stars initially bound to the primary hole are scattered into its tidal disruption loss cone by gravitational interactions with the secondary hole, an enhancement effect that is more pronounced for very unequal-mass binaries; 2) about 25% (40%) of all strongly interacting stars are tidally disrupted by a MBH binary of mass ratio q=1/81 (q=1/243) and eccentricity 0.1; and 3) two mechanisms dominate the fueling of the tidal disruption loss cone, a Kozai non-resonant interaction that causes the secular evolution of the stellar angular momentum in the field of the binary, and the effect of close encounters with the secondary hole that change the stellar orbital parameters in a chaotic way. For a hard MBH binary of 107 \msun and mass ratio 10-2, embedded in an isothermal stellar cusp of velocity dispersion sigma _*=100 km/sec , the tidal disruption rate can be as large as \dot N_* 1 yr-1. This is 4 orders of magnitude higher than estimated for a single MBH fed by two-body relaxation. When applied to the case of a putative intermediate-mass black hole inspiraling onto Sgr A^*, our results predict tidal disruption rates \dot N_* 0.05-0.1 yr-1.