  In this paper, we consider three-dimensional massive gravity's rainbow and
obtain black hole solutions in three different cases of Born-Infeld,
logarithmic, and exponential theories of nonlinear electrodynamics. We discuss
the horizon structure and geometrical properties. Then, we study thermodynamics
of these models by considering the first-order quantum correction effects,
which appears as a logarithmic term in the black hole entropy. We discuss such
effects on the black hole stability and phase transitions. We find that, due to
the quantum corrections, the second-order phase transition happens in
Born-Infeld and logarithmic models. We obtain the modified first law of black
hole thermodynamics in the presence of logarithmic corrections.
