As Torsten Ekedahl observers, this is a modification to fact about finite dimensional algebras, and doesn't concern the statement coproduct on $H$ at all. And as you asked abouthave noted, it's not true as stated for non-semisimple algebras.
However, there is a natural modification, which is true for all finite dimensional Hopf algebras . $A$. Let $X_1,\ldots, X_k$ denote the isomorphism classes of simple objects , of $Rep(A)$, and let $P_1,\ldots P_k$ denote their projective covers. Then we have:
$dim(H) dim(A) = \sum_k (dim X_k) (dim P_k)$.
Of course if $H$ is semi-simple then this recovers the well-known result you mentioned, since $P_k=X_k$ then.
See, for instance, Section 1.47 in the comprehensive lecture notes:
where this is discussed in the context of finite tensor categories (which includes the Hopf algebra example you mentioned). Along the lines of Torsten Ekedahl's comment, this surely depends only on $H$ being an algebra (there is no mention of tensor product...), yet I don't know a reference for that offhand.