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MoTe₂ has recently attracted much attention due to the observation of pressure-induced superconductivity, exotic topological phase transitions, and nonlinear quantum effects. However, there is debate on the intriguing structural phase transitions among various observed phases of MoTe₂, and their connection to the underlying topological electronic properties. In this work, by means of density-functional theory (DFT+U) calculations, we describe the structural phase transition between the polar T_d and nonpolar 1T' phases of MoTe₂ in reference to a hypothetical high-symmetry structure T₀ that exhibits higher-order topological features. We obtain a total of 12 Weyl points, which can be created/annihilated, dynamically manipulated, and switched by tuning a zone-center polar phonon mode in the T_d phase. We also report on the existence of a tunable nonlinear Hall effect in T_d-MoTe₂, and predict an emergent nonlinear surface current under the application of an external electric field. The potential technological applications of the tunable Weyl phase and nonlinear Hall effect will be discussed.