Let $X$ be a minimal compact non-Kähler complex manifold. Suppose that Kodaira dimension $\kappa(X)=0$.

Is there a known example where the canonical bundle is not holomorphically torsion?

For minimal compact Kähler manifolds, the (Kähler extension of the) abundance conjecture predicts that the canonical bundle is holomorphically torsion. Indeed, if $K_X$ is nef, then abundance implies $K_X$ is semi-ample. So we have an Iitaka morphism $\Phi : X \to X_{\text{can}}$, where $\dim(X_{\text{can}})= \kappa(X)=0$, and moreover, $K_X^{\otimes \ell} = f^{\ast} K_{X_{\text{can}}}$, for some $\ell>0$ sufficiently large. Since $X_{\text{can}} = \text{pt}$, we see that $K_X^{\otimes \ell} \simeq f^{\ast} K_{X_{\text{can}}} \simeq \mathcal{O}_X$.

I have never seen the abundance conjecture stated for non-Kähler compact complex manifolds (likely because there is a simple example illustrating its failure which I am unable to bring to mind).

Let $X$ be a minimal compact non-Kähler complex manifold. Suppose that Kodaira dimension $\kappa(X)=0$.

Is there a known example where the canonical bundle is not holomorphically torsion?

For minimal compact Kähler manifolds, the (Kähler extension of the) abundance conjecture predicts that the canonical bundle is holomorphically torsion. Indeed, if $K_X$ is nef, then abundance implies $K_X$ is semi-ample. So we have an Iitaka morphism $\Phi : X \to X_{\text{can}}$, where $\dim(X_{\text{can}})= \kappa(X)=0$, and moreover, $K_X^{\otimes \ell} = f^{\ast} K_{X_{\text{can}}}$, for some $\ell>0$ sufficiently large. Since $X_{\text{can}} = \text{pt}$, we see that $K_X^{\otimes \ell} \simeq f^{\ast} K_{X_{\text{can}}} \simeq \mathcal{O}_X$.

The abundance conjecture fails for non-Kähler compact complex manifolds. Indeed, the abundance conjecture implies the Iitaka $C_{n,m}$ conjecture. Ueno constructed a torus bundle over a torus that violates the $C_{n,m}$ conjecture, and thus violates the abundance conjecture.