  Bearing in mind the application to the collapsar models of gamma-ray bursts
(GRBs), we develop a numerical scheme and code for estimating the deposition of
energy and momentum due to the neutrino pair annihilation ($\nu + {\bar \nu}
\rightarrow e^{-} + e^{+}$) in the vicinity of accretion tori around a Kerr
black hole. Our code is designed to solve the general relativistic neutrino
transfer by a ray-tracing method. To solve the collisional Boltzmann equation
in curved spacetime, we numerically integrate the so-called rendering equation
along the null geodesics. For the neutrino opacity, the charged-current
$\beta$-processes are taken into account, which are dominant in the vicinity of
the accretion tori. The numerical accuracy of the developed code is
certificated by several tests, in which we show comparisons with the
corresponding analytic solutions. Based on the hydrodynamical data in our
collapsar simulation, we estimate the annihilation rates in a post-processing
manner. Increasing the Kerr parameter from 0 to 1, it is found that the general
relativistic effect can increase the local energy deposition rate by about one
order of magnitude, and the net energy deposition rate by several tens of
percents. After the accretion disk settles into a stationary state (typically
later than $\sim 9$ s from the onset of gravitational collapse), we point out
that the neutrino-heating timescale in the vicinity of the polar funnel region
can be shorter than the dynamical timescale. Our results suggest the neutrino
pair annihilation has a potential importance equal to the conventional
magnetohydrodynamic mechanism for igniting the GRB fireballs.
