  We investigate the electronic structure of two Ir$^{5+}$ honeycomb iridates,
Sr$_3$CaIr$_2$O$_9$ and NaIrO$_3$, by means of resonant x-ray techniques. We
confirm that Sr$_3$CaIr$_2$O$_9$ realizes a large spin-orbit driven
non-magnetic $J = 0$ singlet ground state despite sizable tetragonal
distortions of Ir coordinating octahedra. On the other hand, the resonant
inelastic x-ray spectra of NaIrO$_3$ are drastically different from
expectations for a Mott insulator with octahedrally coordinated Ir$^{5+}$. We
find that the data for NaIrO$_3$ can be best interpreted as originating from a
narrow gap non-magnetic $S = 0$ band insulating ground state. Our results
highlight the complex role of the ligand environment in the electronic
structure of honeycomb iridates and the essential role of x-ray spectroscopy to
characterize electronic ground states of insulating materials.
