  We prove that any $\tau$-tilting finite incidence algebra of a finite poset
is representation-finite, and that any $\mathbf{g}$-tame incidence algebra of a
finite simply connected poset is tame. As the converse of these assertions are
known to hold, we obtain characterizations of $\tau$-tilting finite incidence
algebras and $\mathbf{g}$-tame simply connected incidence algebras. Both
results are proved using the theory of concealed algebras. The former will be
deduced from the fact that tame concealed algebras are $\tau$-tilting infinite,
and to prove the latter, we show that wild concealed algebras are not
$\mathbf{g}$-tame. We conjecture that any incidence algebra of a finite poset
is wild if and only if it is not $\mathbf{g}$-tame, and prove a result showing
that there are relatively few possible counterexamples. In the appendix, we
determine the representation type of a $\tau$-tilting reduction of a concealed
algebra of hyperbolic type.
