  We theoretically investigate the electronic band structures and magnetic
properties of ilmenites with edge-sharing IrO$_6$ honeycomb layers, $A$IrO$_3$
with $A=$ Mg, Zn, and Mn, in comparison with a collinear antiferromagnet
MnTiO$_3$. The compounds with $A=$ Mg and Zn were recently reported in
Y.~Haraguchi {\it et al.}, Phys. Rev. Materials {\bf 2}, 054411 (2018), while
MnIrO$_3$ has not been synthesized yet but the honeycomb stacking structure was
elaborated in a superlattice with MnTiO$_3$ in K.~Miura {\it et al.}, Commun.
Mater. {\bf 1}, 55 (2020). We find that, in contrast to MnTiO$_3$, where an
energy gap opens in the Ti $3d$ bands by antiferromagnetic ordering of the
high-spin $S=5/2$ moments, MgIrO$_3$ and ZnIrO$_3$ have a gap in the Ir $5d$
bands under the influence of both spin-orbit coupling and electron correlation.
Their electronic structures are similar to those in the spin-orbit coupled Mott
insulators with the $j_{\rm eff}=1/2$ pseudospin degree of freedom, as found in
monoclinic $A_2$IrO$_3$ with $A=$ Na and Li which have been studied as
candidates for the Kitaev spin liquid. Indeed, we find that the effective
exchange interactions between the $j_{\rm eff}=1/2$ pseudospins are dominated
by the Kitaev-type bond-dependent interaction and the symmetric off-diagonal
interactions. On the other hand, for MnIrO$_3$, we show that the local lattice
structure is largely deformed, and both Mn $3d$ and Ir $5d$ bands appear near
the Fermi level in a complicated manner, which makes the electronic and
magnetic properties qualitatively different from MgIrO$_3$ and ZnIrO$_3$. Our
results indicate that the IrO$_6$ honeycomb network in the ilmenites $A$IrO$_3$
with $A=$ Mg and Zn would offer a good platform for exotic magnetism by the
spin-orbital entangled moments like the Kitaev spin liquid.
