We find an explicit formula of the twisted Reidemeister torsion of the fundamental shadow link complements twisted by the adjoint action of the holonomy representation of the (possibly incomplete) hyperbolic structures, and of the double of hyperbolic polyhedral 3-manifolds twisted by the adjoint action of the holonomy representation of the hyperbolic polyhedral metrics, which turn out to be a product of the determinant of the Gram matrix functions evaluated respectively at the logarithmic holonomies of the meridians and at the edge lengths. As a consequence, we obtain an explicit formula of the twisted Reidemeister torsion of closed hyperbolic 3-manifolds obtained by doing a hyperbolic Dehn-surgery along a fundamental shadow link complement, and of the double of a geometrically triangulated hyperbolic 3-manifold with totally geodesic boundary, respectively in terms of the boundary logarithmic holonomies and of the edge lengths. We notice that by [6] most closed oriented hyperbolic 3-manifolds can be obtained from a suitable fundamental shadow link complement by doing a hyperbolic Dehn-surgery. These formulas play an essential role in the study of the asymptotic expansion of certain quantum invariants in [20].