We investigate the effects of massive gravitons on the rotation curves of the Milky Way, spiral galaxies, and low surface brightness (LSB) galaxies. Using a simple de Rham, Gabadadze, and Tolley (dRGT) massive gravity model, we find a static spherically symmetric metric and a modified Tolman-Oppenheimer-Volkoff (TOV) equation. The dRGT nonlinear graviton interactions generate density and pressures, which behave like dark energy that can mimic the gravitational effects of a dark matter halo. We find that rotation curves of most galaxies can be fitted well by a single constant-gravity parameter γ ∼mg2C ∼10-28 m-1 corresponding to the graviton mass in the range mg∼10-21-10-30 eV depending on the choice of the fiducial metric parameter C ∼1 -1018 m . Fitting the rotation curve of the Milky Way puts a strong constraint on the Yukawa-type coupling of the massive graviton exchange as a result of the shell effects.