Without entropy, there would be no thermodynamics. Heat would not affect the stability of materials, so all materials would be in their ground states (superconductivity would be widespread, but there would be no liquid water to sustain life).
Most of the entropy of materials comes from the vibrations of atoms, and my group has been studying how this vibrational entropy originates with the atomic structures of materials. Our work typically involves neutron inelastic scattering measurements and density functional theory calculations of atom dynamics. Some useful trends have emerged for how vibrational entropy changes with crystal structure or state of chemical order in alloys.
The situation at high temperatures is more complicated, but important because the processing of materials often includes heating. I will describe a hierarchy of non-harmonic behaviors, from quasiharmonic, to anharmonic, to electron-phonon interactions in metals. All have thermodynamic significance. Large anharmonicity alters other properties, too, and can sometimes lead to negative thermal expansion in materials such as ScF3 and Ag2O (these materials shrink when heated).