The intrinsic weak and highly nonlocal dielectric screening of two-dimensional materials is well-known to lead to high sensitivity of their optoelectronic properties to environment. Less studied theoretically is the role of free carriers in those properties. Here, we use ab initio GW and Bethe-Salpeter equation calculations, with a rigorous treatment of dynamical screening and local-field effects, to study the doping dependence of the quasiparticle and optical properties of a monolayer transition-metal dichalcogenide, 2H MoTe₂. We predict a quasiparticle band gap renormalization of several hundreds of meV for experimentally attainable carrier densities and a similarly sizable decrease in the exciton binding energy. This results in an almost constant excitation energy for the lowest-energy exciton resonance with an increasing doping density. Using a newly developed and generally applicable plasmon-pole model and a self-consistent solution of the Bethe-Salpeter equation, we reveal the importance of accurately capturing both dynamical and local-field effects to understand detailed photoluminescence measurements.