I'm just curious as to know if there are any examples in physical chemistry or condensed matter physics where the Dirac equation is preferable to the Shrodinger equation for making predictions on the material at hand?
Graphene is a material that needs the Dirac equation for example. The electron band structure of this material has a closed gap some electrons have "mass=0", that can only we treated with the dirac equation. I dont know if this affects the chemical properties but it sure effects the electric ones.
Yes, relativistic effects become important for reactions involving hydrogen, for instance. There has been a sizable literature in the physical chemistry community on variations of the diagonalization of the Dirac equation (see e. g. the review of Reiher, Wiley Interdiscip. Rev. Comp. Mol. Sci. 2, 139–149, 2012).