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We study topological phase transitions induced by optical phonon-mode lattice distortions in ZrTe₅, a layered van der Waals topological insulating material, by using first-principles and effective Hamiltonian methods. We find that five of the six zone-centered A_g Raman phonon modes, which protect the crystalline symmetries, induce transition from strong to weak topological insulator with Dirac semimetallic state emerging at the transition point. Moreover, all zone-centered infra-red phonon modes induce transition from topological insulating to the Weyl semimetallic phase by breaking the global inversion symmetry. Thus achieved Weyl phases are robust, tunable and in close proximity to the Fermi level.The experimental implications of our results are discussed.