Let $X$ be a compact complex manifold. Any global holomorphic $k$-form $\omega \in H^0(X, \Omega^k)$ is automatically harmonic with respect to any choice of hermitian structure on $X$. When $X$ is Kähler, by the Kähler identities, this implies that $\mathrm{d}{\omega} = 0$.
I am wondering if the same is true when $X$ is not Kähler. If not, what is an example of a nonclosed global holomorphic $k$-form.
This happens on the Iwasawa manifold.
Let $G$ be the Heisenberg group of complex upper-triangular $3 \times 3$ matrices $$ G = \begin{pmatrix} 1 & x & y \\ 0 & 1 & z \\ 0 & 0 & 1 \end{pmatrix} $$ and let $\Gamma \subset G$ be the discrete subgroup generated by matrices whose non-diagonal entries are Gaussian integers. The Iwasawa manifold is the quotient $X := G / \Gamma$. It is a compact complex manifold.
There are three linearly independent holomorphic forms on $X$: $\alpha = dx$, $\beta = dy$ and $\gamma = dz - x \, dy$. They're defined by looking at the entries of $M^{-1} \, dM$ for $M(x,y,z) \in G$.
We see that $d \gamma = - \alpha \wedge \beta \not= 0$, so $\gamma$ is a non-closed holomorphic form. In particular, this proves $X$ is not Kahler.
The facts here are pretty much verbatim from Michel Schweitzer's manuscript Autour de la cohomologie de Bott-Chern.