  We characterize the toric face rings that are normal (respectively
seminormal). Extending results about local cohomology of Brun, Bruns, Ichim, Li
and R\"omer of seminormal monoid rings and Stanley toric face rings, we prove
the vanishing of certain graded parts of local cohomology of seminormal toric
face rings. The combinatorial formula we obtain generalizes Hochster's formula.
We also characterize all (necessarily seminormal) toric face rings that are
$F$-pure or $F$-split over a field of characteristic $p>0$. An example is given
to show that $F$-injectivity does not behave well with respect to face
projections of toric face rings. Finally, it is shown that weakly $F$-regular
toric face rings are normal affine monoid rings.
