We investigate the impact of thermal effects in binary neutron star (BNS) merger remnants and their imprints on gravitational waves (GWs) through fully general relativistic simulation using either a Γ−law or a fully tabulated equations of state approach to allow for shock heating. We consider irrotational neutron stars in quasi-circular orbits that merger and form a transient hypermassive neutron star (HMNS) that lasts over 150 ms after merger. Our simulations reveal significant differences in the physical properties of the HMNS such as rotation profiles, matter oscillation modes and internal temperature, due to these two approaches which in turn modify the GW peak frequencies and so lead to distinct quasi-universal relations. Besides, we explore if these effects can be detectable by third-generation gravitational-wave detectors like the Cosmic Explorers and the Einstein Telescope through a Bayesian selection model. We find that these differences are detectable at significant distances, with variances dependent on the equation of state and source inclination. These findings highlight the necessity of a self-consistent treatment of thermal effects for precise GW astronomy.