What do lattice theorists do?
They study the lattice vibrations, surface interface, electrical and magnetic properties of crystals.
They can also suggest mechanisms and ways to improve certain crystal properties.
The biggest topic in material science is probably superconductivity. As far as I can tell, most high temperature superconductors are crystals. Superconductivity works at a quantum level as well as the classical level. Quantum level because we're talking about the BCS theory (named after John Bardeen, Leon Cooper and J. Robert Schrief) and the Cooper pair (electron-phonon interaction). Classical because the thermal properties of superconductors are different than normal material.
Another one is studying directly the phonon of a crystal (crystal vibrations). This is especially useful in electronics because one major set back in electronics besides computing power is the overheating problem. Overheating reduces lifetime of our devices and studying crystal vibrations can help improve them.