  We predict a quantum phase transition from normal to topological insulators
in the 5$d$ transition metal oxide Na$_2$IrO$_3$, where the transition can be
driven by the change of the long-range hopping and trigonal crystal field
terms. From the first-principles-derived tight-binding Hamiltonian we determine
the phase boundary through the parity analysis. In addition, our
first-principles calculations for Na$_2$IrO$_3$ model structures show that the
interlayer distance can be an important parameter for the existence of a
three-dimensional strong topological insulator phase. Na$_2$IrO$_3$ is
suggested to be a candidate material which can have both a nontrivial topology
of bands and strong electron correlations.
