In this paper, we first extract general uncharged accelerating BTZ black hole solutions and study some of their properties. Our analysis shows that spacetime's asymptotical behavior depends on four parameters (the cosmological constant, mass, the acceleration parameter, and the topological constant). Then, we study the temperature of these black holes and find a divergence point, which this point reveals a phase transition between physical and non-physical black holes. Next, we extend our study for extracting charged accelerating BTZ black hole solutions in the presence of a nonlinear electrodynamics field, known as conformally invariant Maxwell. Our findings indicate that there is a coupling between the electrical charge and other quantities of the accelerating BTZ black holes. We find that the asymptotical behavior of charged accelerating BTZ black holes depends on six parameters (the cosmological constant, the electrical charge, mass, the acceleration parameter, topological constant, and angular coordinate). Finally, we investigate the effects of various parameters on the temperature of these black holes. There are two exciting behaviors for the temperature by considering different cosmological constant values. A divergence point exists for the temperature when the cosmological constant is negative. However, for the positive value of the cosmological constant, there are three critical points in the temperature; two roots and a divergence point. These points give us information about phase transitions between physical and non-physical black holes.