  Topological superconductors should be able to provide essential ingredients
for quantum computing, but are very challenging to realize. Spin-orbit
interaction in iron-based superconductors opens the energy gap between the
$p$-states of pnictogen and $d$-states of iron very close to the Fermi level,
and such $p$-states have been recently experimentally detected. Density
functional theory predicts existence of topological surface states within this
gap in FeTe$_{1-x}$Se$_x$ making it an attractive candidate material. Here we
use synchrotron-based angle-resolved photoemission spectroscopy and band
structure calculations to demonstrate that FeTe$_{1-x}$Se$_x$ (x=0.45) is a
superconducting 3D Dirac semimetal hosting type-I and type-II Dirac points and
that its electronic structure remains topologically trivial. We show that the
inverted band gap in FeTe$_{1-x}$Se$_x$ can possibly be realized by further
increase of Te content, but strong correlations reduce it to a sub-meV size,
making the experimental detection of this gap and corresponding topological
surface states very challenging, not to mention exact matching with the Fermi
level. On the other hand, the $p-d$ and $d-d$ interactions are responsible for
the formation of extremely flat band at the Fermi level pointing to its
intimate relation with the mechanism of high-T$_c$ superconductivity in IBS.
