Monolayer transition metal dichalcogenide (TMDC) semiconductors exhibit strong excitonic optical resonances, which serve as a microscopic, noninvasive probe into their fundamental properties. Like the hydrogen atom, such excitons can exhibit an entire Rydberg series of resonances. Excitons have been extensively studied in most TMDCs (MoS₂, MoSe₂, WS₂, and WSe₂), but detailed exploration of excitonic phenomena has been lacking in the important TMDC material molybdenum ditelluride (MoTe₂). Here, we report an experimental investigation of excitonic luminescence properties of monolayer MoTe₂ to understand the excitonic Rydberg series, up to 3s. We report a significant modification of emission energies with temperature (4 to 300 K), thereby quantifying the exciton-phonon coupling. Furthermore, we observe a strongly gate-tunable exciton-trion interplay for all the Rydberg states governed mainly by free-carrier screening, Pauli blocking, and band gap renormalization in agreement with the results of first-principles GW plus Bethe-Salpeter equation approach calculations. Our results help bring monolayer MoTe₂ closer to its potential applications in near-infrared optoelectronics and photonic devices.