There are many compact examples that are not Kähler. For example, by a theorem of Borel, every complex semisimple Lie group $G$ contains a discrete, cocompact subgroup $\Gamma\subset G$. If you let $X = G/\Gamma$, then $X$ will be a holomorphically parallelizable compact complex manifold. The ring of right-invariant holomorphic differential forms on $G$, say, $R$, descends to be a ring (still called $R$) of holomorphic forms on $X$. This ring is closed under exterior derivative, of course, and every holomorphic differential form on $X$ belongs to $R$.

Now, taking different examples of $G$ and $\Gamma$, one can construct many examples of holomorphic forms on compact manifolds that are not closed, are closed but not exact, etc. Each nonzero form in $R$ has empty vanishing locus.

Nontrivial Kähler examples can be constructed this way: Let $Z$ be a compact Kähler manifold of complex dimension $2n$ that supports a holomorphic symplectic $2$-form $\Omega$, and let $X\subset Z$ be a smooth subvariety of dimension $d>n$. Then the pullback of $\Omega$ to $X$ cannot vanish anywhere on $X$ for dimension reasons, but there is no reason to believe (except when $d=2n{-}1$) that the rank of this pullback needs to be constant. $X$ will be Kähler, but, if the rank of the pullback is not constant, then you won't be able to write $X$ as a product of Kähler manifolds, some of which support the holomorphic $2$-form as a symplectic form.

compact. Otherwise take an arbitrary manifold with an arbitrary non-zero (i.e., not constant zero) 2-form and consider the open set where it is not zero. $\endgroup$ – Sándor Kovács Apr 12 '13 at 15:21