  In this paper, we report on the $\tau$-tilting finiteness of some classes of
finite-dimensional algebras over an algebraically closed field, including
symmetric algebras of polynomial growth, $0$-Hecke algebras, and $0$-Schur
algebras. Consequently, we find that derived equivalence preserves the
$\tau$-tilting finiteness over symmetric algebras of polynomial growth, and
self-injective cellular algebras of polynomial growth are $\tau$-tilting
finite. Furthermore, the representation-finiteness and $\tau$-tilting
finiteness over 0-Hecke algebras and 0-Schur algebras (with few exceptions)
coincide.
