  In this paper, we investigate the thermodynamics of dyonic black holes in the
presence of Born-Infeld electromagnetic field. We show that electric-magnetic
duality reported for dyonic solutions with Maxwell field is omitted in case of
Born-Infeld generalization. We also confirm that generalization to nonlinear
field provides the possibility of canceling the effects of cosmological
constant. This is done for nonlinearity parameter with $10^{-33}\mbox{ eV}^{2}$
order of magnitude which is high nonlinearity regime. In addition, we show that
for small electric/magnetic charge and high nonlinearity regime, black holes
would develop critical behavior and several phases. In contrast, for highly
charged case and Maxwell limits (small nonlinearity), black holes have one
thermal stable phase. We also find that the pressure of the cold black holes is
bounded by some constraints on its volume while hot black holes' pressure has
physical behavior for any volume. In addition, we report on possibility of
existences of triple point and reentrant of phase transition in thermodynamics
of these black holes. Finally, We show that if electric and magnetic charges
are identical, the behavior of our solutions would be Maxwell like (independent
of nonlinear parameter and field). In other words, nonlinearity of
electromagnetic field becomes evident only when these black holes are charged
magnetically and electrically different.
