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Motivated by recent developments of black hole thermodynamics in de Rham, Gabadadze, and Tolley (dRGT) massive gravity, we study the critical behaviors of topological Anti-de Sitter (AdS) black holes in the presence of Born-Infeld nonlinear electrodynamics. Here the cosmological constant appears as a dynamical pressure of the system and its corresponding conjugate quantity is interpreted as thermodynamic volume. This shows that, besides the Van der Waals-like SBH/LBH phase transitions, the so-called reentrant phase transition (RPT) appears in four-dimensional space-time when the coupling coefficients cim2 of massive potential and Born-Infeld parameter b satisfy some certain conditions. In addition, we also find the triple critical points and the small/intermediate/large black hole phase transitions for d=5.