Recent first detections of the cross-correlation of the thermal Sunyaev-Zeldovich (tSZ) signal in Planck cosmic microwave background (CMB) temperature maps with gravitational lensing maps inferred from the Planck CMB data and the CFHTLenS galaxy survey provide new probes of the relationship between baryons and dark matter. Using cosmological hydrodynamics simulations, we show that these cross-correlation signals are dominated by contributions from hot gas in the intracluster medium (ICM), rather than diffuse, unbound gas located beyond the virial radius (the missing baryons). Thus, these cross-correlations offer a tool with which to study the ICM over a wide range of halo masses and redshifts. In particular, we show that the tSZ - CMB lensing cross-correlation is more sensitive to gas in lower-mass, higher-redshift halos and gas at larger cluster-centric radii than the tSZ - galaxy lensing cross-correlation. Combining these measurements with primary CMB data will constrain feedback models through their signatures in the ICM pressure profile. We forecast the ability of ongoing and future experiments to constrain such ICM parameters, including the mean amplitude of the pressure - mass relation, the redshift evolution of this amplitude, and the mean outer logarithmic slope of the pressure profile. The results are promising, with $approx 5-20$% precision constraints achievable with upcoming experiments, even after marginalizing over cosmological parameters.
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