  The thermal Sunyaev-Zel'dovich (tSZ) effect induces a Compton-$y$ distortion
in cosmic microwave background (CMB) temperature maps that is sensitive to a
line of sight integral of the ionized gas pressure. By correlating the
positions of galaxies with maps of the Compton-$y$ distortion, one can probe
baryonic feedback processes and study the thermodynamic properties of a
significant fraction of the gas in the Universe. Using a model fitting
approach, we forecast how well future galaxy and CMB surveys will be able to
measure these correlations, and show that powerful constraints on halo pressure
profiles can be obtained. Our forecasts are focused on correlations between
galaxies and halos identified by the upcoming Dark Energy Spectroscopic
Instrument survey and tSZ maps from the Simons Observatory and CMB-S4
experiments, but have general applicability to other surveys, such as the Large
Synoptic Survey Telescope. We include prescriptions for observational
systematics, such as halo miscentering and halo mass bias, demonstrating
several important degeneracies with pressure profile parameters. Assuming
modest priors on these systematics, we find that measurements of halo-$y$ and
galaxy-$y$ correlations with future surveys will yield tight constraints on the
pressure profiles of group-scale dark matter halos, and enable current feedback
models to either be confirmed or ruled out.
