Context. We investigate the deposition of energy and momentum due to the annihilation of neutrinos (ν) and antineutrinos (¯ν) in the vicinity of steady, axisymmetric accretion tori around stellar-mass black holes (BHs). This process is widely considered as an energy source for driving ultrarelativistic outflows with the potential to produce gamma-ray bursts. Aims. We analyze the influence of general relativistic (GR) effects in combination with different neutrinosphere properties on the ν¯ν-annihilation efficiency and spatial distribution of the energy deposition rate. Methods. Assuming axial symmetry, we numerically compute the annihilation rate 4-vector. For this purpose, we construct the local neutrino distribution by ray-tracing neutrino trajectories in a Kerr space-time using null geodesics. We vary the value of the dimensionless specific angular momentum a of the central BH, which provides the gravitational field in our models. We also study different shapes of the neutrinospheres, spheres, thin disks, and thick accretion tori, whose structure ranges from idealized tori to equilibrium non-selfgravitating matter distributions. Furthermore, we compute Newtonian models where the influence of the gravitational field on the annihilation process is neglected. Results. Compared to Newtonian calculations, GR effects increase the total annihilation rate measured by an observer at infinity by a factor of two when the neutrinosphere is a thin disk, but the increase is only ≈25% for toroidal and spherical neutrinospheres. Comparing cases with similar luminosities, thin disk models yield the highest energy deposition rates by ν¯ν-annihilation, and spherical neutrinospheres the lowest ones, independently of whether GR effects are included. Increasing a from 0 to 1 enhances the energy deposition rate measured by an observer at infinity by roughly a factor of 2 due to the change of the inner radius of the neutrinosphere. General relativity and rotation cause important differences in the spatial distribution of the energy deposition rate by ν¯ν-annihilation.