Here is a straightforward example with $\Delta \neq 0$. I think it can be adapted to give an example without a boundary.

Let $X_0 = C \times C'$ be a product of elliptic curves viewed as an elliptic fibration $X_0 \to C$ with a fixed section $S_0$ and fiber $E_0$. Let $\mu : X \to X_0$ be the blowup of the intersection point $p = S_0 \cap E_0$. I'll denote the exceptional divisor by $A$ and the two strict tranforms by $S$ and $E$.

Now consider the log canonical pair $(X, \Delta = S + F + E)$ where $F$ is the strict transform of a fiber disjoint from $p$. We have that $K_X \sim_\mathbb{Q} A$ and
$$
L = K_X + \Delta = K_X + S + F + E
$$
is big, nef and even semiample. $L$ is zero precisely on $E$ so the basepoint free linear series $|mL|$ for $m \gg 0$ induces a log canonical contraction $f : X \to Y$ that contracts $E$ to an elliptic singularity and $Y$ is the log canonical model of $(X, \Delta)$.

Now we want to show that $\mathcal{O}_X(mL) = \omega_X(\Delta) \otimes \mathcal{O}_X((m-1)L)$ has nonzero higher cohomology. First note that $f_*\mathcal{O}_X(mL) = \mathcal{O}_Y(m(K_X + f_*\Delta))$ is sufficiently ample on $Y$ so $H^j(f_*\mathcal{O}_X(mL)) = 0$ for $j > 0$. Therefore
$$
H^1(X, \mathcal{O}_X(mL)) = H^0(Y, R^1f_*\mathcal{O}_X(mL))
$$
from Leray spectral sequence. Now $R^1f_*\mathcal{O}_X(mL)$ is a skyscraper sheaf at $f(E)$ and we can compute this is nonzero by the theorem on formal functions using: $(1)$ $E$ is an elliptic curve, $(2)$ $mL|_E = \mathcal{O}_E$.

Here the restriction of $mL$ to $E$ contributes to the cohomology since $E$ is a log canonical center of the pair. If I understand correctly, this is the only way that KV vanishing can fail for log canonical pairs:

$\textbf{Theorem:}$ Let $(X, \Delta)$ be a projective log canonical pair. Let $L$ be a big and nef $\mathbb{Q}$-Cartier divisor such that $L$ remains big when restricted to each log canonical center of $(X,\Delta)$. Then
$$
H^j(X, \mathcal{O}_X(K_X + \Delta + L)) = 0
$$
for $j > 0$.

This is a special cause of Theorem 1.10 in this paper of Fujino. See also Theorem 1.7 in the paper linked in the question.