  Motivated by recent developments of BTZ black holes and interesting results
of massive gravity, we investigate massive BTZ black holes in the presence of
Maxwell and Born-Infeld (BI) electrodynamics. We study geometrical properties
such as type of singularity and asymptotical behavior as well as thermodynamic
structure of the solutions through canonical ensemble. We show that despite the
existence of massive term, obtained solutions are asymptotically (a)dS and have
a curvature singularity at the origin. Then, we regard varying cosmological
constant and examine the Van der Waals like behavior of the solutions in
extended phase space. In addition, we employ geometrical thermodynamic
approaches and show that using Weinhold, Ruppeiner and Quevedo metrics leads to
existence of ensemble dependency while HPEM metric yields consistent picture.
For neutral solutions, it will be shown that generalization to massive gravity
leads to the presence of non-zero temperature and heat capacity for vanishing
horizon radius. Such behavior is not observed for linearly charged solutions
while generalization to nonlinearly one recovers this property.
