The Transiting Exoplanet Survey Satellite (TESS) will conduct a two-year wide-field survey searching for transiting planets around bright stars. Many TESS discoveries will be amenable to mass characterization via ground-based radial velocity measurements with any of a growing suite of existing and anticipated velocimeters in the optical and near-infrared. In this study we present an analytical formalism to compute the number of radial velocity (RV) measurements—and hence the total observing time—required to characterize RV planet masses with the inclusion of either a white or correlated noise activity model. We use our model to calculate the total observing time required to measure all TESS planet masses from the expected TESS planet yield while relying on our current understanding of the targeted stars, stellar activity, and populations of unseen planets that inform the expected RV precision. We also present specialized calculations applicable to a variety of interesting subsets of TESS planets including the characterization of 50 planets smaller than 4 Earth radii, which is expected to take as little as 60 nights of observation. However, the efficient RV characterization of such planets requires a priori knowledge of the “best” targets, which we argue can be identified prior to the conclusion of the TESS planet search based on our calculations. Our results highlight the comparable performance of optical and near-IR spectrographs for most planet populations except for Earths and temperate TESS planets, which are more efficiently characterized in the near-IR. Lastly, we present an online tool to the community to compute the total observing times required to detect any transiting planet using a user-defined spectrograph (RVFC; http://maestria.astro.umontreal.ca/rvfc).