For molecular orbitals, how is an energy of zero set or defined?
The answer technically depends on the method. In most cases, meaning HF or DFT with standard basis sets, it is infinite separation of all bare nuclei and all electrons. One exception known to me is VASP (a periodic, solid-state code), which uses pseudopotentials to avoid having to model the core singularity and core electrons. There, the zero-point is in reference to some atomic state.

Note that the MO energies are basis-set dependent, which is most pronounced and most easily understood for core electrons.