The Roothaan-Hartree-Fock (HF) method has been implemented in deMon-DynaRho within the resolution-of-the-identity (RI) auxiliary-function approximation. While previous studies have focused primarily upon the effect of the RI approximation on total energies, very little information has been available regarding the effect of the RI approximation on orbital energies, even though orbital energies play a central role in many theories of ionization and excitation. We fill this gap by testing the accuracy of the RI approximation against non-RI-HF calculations using the same basis sets, for the occupied orbital energies and an equal number of unoccupied orbital energies of five small molecules, namely CO, N-2, CH2O, C2H4, and pyridine (in total 102 orbitals). These molecules have well-characterized excited states and so are commonly used to test and validate molecular excitation spectra computations. Of the deMon auxiliary basis sets tested, the best results are obtained with the (44) auxiliary basis sets, yielding orbital energies to within 0.05 eV, which is adequate for analyzing typical low resolution polyatomic molecule ionization and excitation spectra. Interestingly, we find that the error in orbital energies due to the RI approximation does not seem to increase with the number of electrons. The absolute RI error in the orbital energies is also roughly related to their absolute magnitude, being larger for the core orbitals where the magnitude of orbital energy is large and smallest where the molecular orbital energy is smallest. Two further approximations were also considered, namely uniterated ("zero-order") and single-iteration ("first-order") calculations of orbital energies beginning with a local density approximation initial guess. We find that zero- and first-order orbital energies are very similar for occupied but not for unoccupied orbitals, and that the first-order orbital energies are fairly close to the corresponding fully converged values. Typical root mean square errors for first-order calculations of orbital energies are about 0.5 eV for occupied and 0.05 eV for unoccupied orbitals. Also reported are a few tests of the effect of the RI approximation on total energies using deMon basis sets, although this was not the primary objective of the present work.