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We theoretically study the topological properties of magnons and the relevant magnon thermal Hall effect in trimerized Lieb lattice ferromagnets in the presence of next-nearest-neighbor Dzyaloshinskii-Moriya interactions. By calculating the magnon band structures and their Chern numbers with the linear spin-wave theory, we show that the system can undergo a phase transition between a magnonic topological insulator phase and a magnonic trivial insulator phase. The main results are presented in the form of topological phase diagrams, where the Chern numbers or magnon thermal Hall conductivity are shown as a function of the two lattice trimerization parameters. We find a sharp change of the thermal Hall conductivity across the critical point of phase transformations associated with topological nontrivial edge states. The behaviors reflect that the existence of trimerizations breaks the C4 rotational symmetry of the Lieb lattice. Finally, we show that our theoretical predictions could be experimentally realized in high-temperature cuprate superconductors or organic magnetic materials. © 2024 authors. Published by the American Physical Society. Published by the American Physical Society under the terms of the 'https://creativecommons.org/licenses/by/4.0/'Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Open access publication funded by King Abdullah University of Science and Technology.