Two of Transiting Exoplanet Survey Satellite's major science goals are to measure masses for 50 planets smaller than 4 Earth radii and to discover high-quality targets for atmospheric characterization efforts. It is important that these two goals are linked by quantifying what precision of mass constraint is required to yield robust atmospheric properties of planets. Here, we address this by conducting retrievals on simulated James Webb Space Telescope transmission spectra under various assumptions for the degree of uncertainty in the planets mass for a representative population of seven planets ranging from terrestrials to warm Neptunes to hot Jupiters. Only for the cloud-free, low-metallicity gas giants are we able to infer exoplanet mass from transmission spectroscopy alone, to ∼10% accuracy. For low-metallicity cases (<4× solar) we are able to accurately constrain atmospheric properties without prior knowledge of the planet's mass. For all other cases (including terrestrial-like planets), atmospheric properties can only be inferred with a mass precision of better than 50%. At this level, though, the widths of the posterior distributions of the atmospheric properties are dominated by the uncertainties in mass. With a precision of 20%, the widths of the posterior distributions are dominated by the spectroscopic data quality. Therefore, as a rule of thumb, we recommend a 50% mass precision for initial atmospheric characterization and a 20% mass precision for more detailed atmospheric analyses.