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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, Tc. By showing that the Chern numbers of the magnonic bands are distinct above and below Tc, 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 Tc, which can serve as an experimental probe of its topological nature. Our theory predicts that in CrI3 such a phenomenon exists and is experimentally accessible. [ABSTRACT FROM AUTHOR]