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Monolayers of transition-metal dichalcogenides (TMDs) hold great promise as future nanoelectronic and optoelectronic devices. An essential feature for achieving high device performance is the use of suitable supporting substrates, which can affect the electronic and optical properties of these two-dimensional (2D) materials. Here, we perform many-body GW calculations using the SternheimerGW method to investigate the quasiparticle band structure of monolayer MoS2 subject to an effective dielectric screening model, which is meant to approximately describe substrate polarization in real device applications. We show that, within this model, the dielectric screening has a sizable effect on the quasiparticle band gap; for example, the gap renormalization is as large as 250 meV for MoS2 with model screening corresponding to SiO2. Within the G0W0 approximation, we also find that the inclusion of the effective screening induces a direct band gap, in contrast to the unscreened monolayer. We also find that the dielectric screening induces an enhancement of the carrier effective masses by as much as 27% for holes, shifts plasmon satellites, and redistributes quasiparticle weight. Our results highlight the importance of the dielectric environment in the design of 2D TMD-based devices.