Nonadiabatic coupling between electrons and molecular motion at metal surfaces leads to energy dissipation and dynamic steering effects during chemical surface dynamics. We present a theoretical approach to the scattering of molecules from metal surfaces that incorporates all nonadiabatic and quantum nuclear effects due to the coupling of the molecular degrees of freedom to the electrons in the metal. This is achieved with the hierarchical equations of motion (HEOM) approach, combined with a matrix product state representation in twin space. The method is applied to the scattering of nitric oxide from Au(111), for which strongly nonadiabatic energy loss during scattering has been experimentally observed, thus presenting a significant theoretical challenge. Since the HEOM approach treats the molecule-surface coupling exactly, it captures the interplay between nonadiabatic and quantum nuclear effects. Finally, the data obtained by the HEOM approach are used as a rigorous benchmark to assess various mixed quantum-classical methods, from which we derive insights into the mechanisms of energy dissipation and the suitable working regimes of each method.