A general (logarithmic) Kodaira-Akizuki-Nakano vanishing theorem in characteristic $p$ is proven in $\S$ 11 of

<cite authors="Esnault, Hélène; Viehweg, Eckart">Esnault, Hélène; Viehweg, Eckart: *Lectures on vanishing theorems*. Notes, grew out of the DMV-seminar on algebraic geometry, held at Reisensburg, October 13-19, 1991, DMV Seminar. 20. Basel: Birkhäuser Verlag. 164 p. (1992). [ZBL0779.14003](https://zbmath.org/?q=an:0779.14003)</cite>,

see in particular Corollary 11.3. The precise statement is the following

 >**Theorem.** Let $k$ be a perfect field, let $X$ be a proper smooth $k$-scheme and $D \subset X$ a normal crossing divisor, both admitting a lifting $\tilde{D} \subset \tilde{X}$ to $W_2(k)$. 

>For $p+q < \mathrm{min}\{\mathrm{char}(k), \, \dim X\}$ and $\mathscr{L}$ ample and invertible, we have $$H^q(X, \, \Omega^p(\log D) \otimes \mathscr{L}^{-1})=0.$$

So you cannot have a counterexample to $(*)$, unless $k$ is not perfect or the pair $(X, \, D)$ do not lift to the ring of Witt vectors. I guess that counterexamples in the last case could be obtained by using the same strategy of Reynaud's famous counterexample for Kodaira vanishing in positive characteristic, but unfortunately I do not know any of them