We present a microscopic study of the nuclear magnetic relaxation rate 1/T1 based on the fiveorbital model for iron-based superconductors. We mainly discuss the effect of the "inelastic" quasiparticle damping rate γ due to many-body interaction on the size of the coherence peak, for both s++ and s±-wave superconducting states. We focus on Ba(Fe1−xCox)2As2, and systematically evaluate γ in the normal state from the experimental resistivity, from optimally to over-doped compounds. Next, γ in the superconducting state is calculated microscopically based on the second order perturbation theory. In optimally doped compounds (Tc ∼ 30 K), it is revealed that the coherence peak on 1/T1T is completely suppressed due to large γ for both s++ and s±-wave states. On the other hand, in heavily over doped compounds with Tc < 10 K, the coherence peak could appear for both pairing states, since γ at Tc is quickly suppressed in proportion to T 2 c . By making careful comparison between theoretical and experimental results, we conclude that it is difficult to discriminate between s++ and s±-wave states from the present experimental results.