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We characterize the toric face rings that are normal (respectivelyseminormal). Extending results about local cohomology of Brun, Bruns, Ichim, Liand R\"omer of seminormal monoid rings and Stanley toric face rings, we provethe vanishing of certain graded parts of local cohomology of seminormal toricface 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 givento show that $F$-injectivity does not behave well with respect to faceprojections of toric face rings. Finally, it is shown that weakly $F$-regulartoric face rings are normal affine monoid rings.

Seminormality and local cohomology of toric face rings