  The study of the magnonic thermal Hall effect in magnets with
Dzyaloshinskii-Moriya interaction (DMI) has recently drawn attention because of
the underlying topology. Topological phase transitions may arise when there
exist two or more distinct topological phases, and they are often revealed by a
gap-closing phenomenon. In this work, we consider the magnons in honeycomb
ferromagnets described by a Heisenberg Hamiltonian containing both an
out-of-plane DMI and a Zeeman interaction. We demonstrate that the magnonic
system exhibits temperature (or magnetic field) driven topological phase
transitions due to magnon-magnon interactions. Specifically, when the
temperature increases, the magnonic energy gap at Dirac points closes and
reopens at a critical temperature, $T_c$. By showing that the Chern numbers of
the magnonic bands are distinct above and below $T_c$, we confirm that the
gap-closing phenomenon is indeed a signature for the topological phase
transitions. Furthermore, our analysis indicates that the thermal Hall
conductivity in the magnonic system exhibits a sign reversal at $T_c$, which
can serve as an experimental probe of its topological nature. Our theory
predicts that in $\rm{CrI_3}$ such a phenomenon exists and is experimentally
accessible.
