Motivated by the recent experimental realization of twisted transition metal dichalcogenide bilayers, we study a simplified model driven by different forms of monochromatic light. As a concrete and representative example we use parameters that correspond to a twisted MoTe$_2$ homobilayer. First, we consider irradiation with circularly polarized light in free space and demonstrate that the corresponding Floquet Hamiltonian takes the same form as the static Hamiltonian, only with a constant overall shift in quasi-energy. This is in stark contrast to twisted bilayer graphene, where new terms are typically generated under an analagous drive. Longitudinal light, on the other hand, which can be generated from the transverse magnetic mode in a waveguide, has a much more dramatic effect--it renormalizes the tunneling strength between the layers, which effectively permits the tuning of the twist angle {em in-situ}. We find that, by varying the frequency and amplitude of the drive, one can induce a topological transition that cannot be obtained with the traditional form of the Floquet drive in free space. Furthermore, we find that strong drives can have a profound effect on the layer pseudospin texture of the twisted system, which coincides with multiple simultaneous band gap closings in the infinite-frequency limit. Surprisingly, these bandgap closings are not associated with topological transitions. For high but finite drive frequencies near $0.7$eV, the infinite-frequency band crossings become band gap minima of the order of $10^{-6}$ eV or smaller.
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