In molecular devices, electronic motion is coupled to vibrational modes of the molecule. As such, these devices offer an outstanding opportunity to study fundamental aspects of the coupling between electronic and bosonic degrees of freedom. In this talk we consider one such example: the nonequilibrium heat exchange between a conduction band and a bosonic bath mediated by a single molecule. Under suitable resonance conditions one can map a variety of molecular devices onto an effective model which is exactly solvable. In this model we calculate the heat conductance between the bosonic bath and the electronic conduction band for any finite temperature difference, revealing surprising results. For example: for an ohmic bath we find that the heat current initially depends in a quartic manner on the temperature difference, crossing over to a linear dependence when the temperatures involved are large compared to the phononic resonance width. We discuss the origin of this phenomenon which markedly differs from the behavior found in purely fermionic systems.