  Hyperkagome iridate, Na$_4$Ir$_3$O$_8$, has been regarded as a promising
candidate material for a three-dimensional quantum spin liquid. Here the
three-dimensional network of corner-sharing triangles forms the hyperkagome
lattice of Ir$^{4+}$ ions. Due to strong spin-orbit coupling, the local moments
of Ir$^{4+}$ ions are described by the pseudospin $j_{\rm eff} = 1/2$ Kramers
doublet. The Heisenberg model on this lattice is highly frustrated and
quantum/classical versions have been studied in earlier literature. In this
work, we derive a generic local-moment model beyond the Heisenberg limit for
the hyperkagome iridate by considering multi-orbital interactions for all the
$t_{2g}$ orbitals and spin-orbit coupling. The lifting of massive classical
degeneracy in the Heisenberg model by various spin-anisotropy terms is
investigated at the classical level and the resulting phase diagram is
presented. We find that different anisotropy terms prefer distinct classes of
magnetically ordered phases, often with various discrete degeneracy. The
implications of our results for recent $\mu$SR and NMR experiments on this
material and possible quantum spin liquid phases are discussed.
