Binary neutron star mergers provide a unique probe of the dense-matter equation of state (EOS) across a wide range of parameter space, from the cold EOS during the inspiral to the finite-temperature EOS following the merger. In this talk, I will start with an overview of what we have learned about the EOS from the first few LIGO-Virgo observations of binary neutron star inspirals. I will then discuss what additional EOS information can be extracted by studying the late-stages of a binary neutron star merger, during which time the matter is heated to significant temperatures and can deviate away from its initial equilibrium composition. I will present a new set of neutron star merger simulations, which use a parametrized framework for calculating the EOS at arbitrary temperatures and compositions. I will show how varying the properties of the particle effective mass affects the thermal profile of the post-merger remnant and how this, in turn, and influences the post-merger evolution. Finally, I will discuss the imprint of the nuclear symmetry energy on the post-merger properties and I will comment on the prospects for constraining these nuclear parameters with future gravitational wave events.