EDIT: New example, hopefully this one works.

Blow up $\mathbb{P}^2$ at a point $p$, then blow up the resulting surface at a point $q$ on the exceptional divisor. The resulting surface has Picard group generated by the class $H$ of a line, the proper transform $E_1$ of the first exceptional divisor, and the second exceptional divisor $E_2$. We have $E_1^2 = -2$ and $E_2^2 = -1$, while $E_1\cdot E_2 = 1$. Now consider $D = E_1 +2E_2$. The Zariski decomposition of $D$ is $P = 0$, $N = E_1+2E_2$, as no subdivisor effective divisor supported on $E_1$ and $E_2$ is nef. Then $D\cdot E_1 = 0$. We can get strict inequality instead by taking $D = E_1+3E_2$, for instance.

Let $X\to \mathbb{P}^2$ be the blowup of

Blow up $\mathbb{P}^2$ at one a point $p$. Let $D$ be p$, then blow up the full preimage of resulting surface at a line in $\mathbb{P}^2$ passing through the point $p$, consisting of q$ on the exceptional divisor. The resulting surface has Picard group generated by the class $E$ and H$ of a line, the proper transform $H-E$ E_1$ of the line through first exceptional divisor, and the second exceptional divisor $p$. E_2$. We have $E_1^2 = -2$ and $E_2^2 = -1$, while $E_1\cdot E_2 = 1$. Now consider $D = E_1 +2E_2$. The Zariski decomposition of $D$ is $D P = (H-E) + E$0$, and $N = E_1+2E_2$, as no subdivisor is nef. Then $D\cdot E=0$E_1 = 0$.

Let $X\to \mathbb{P}^2$ be the blowup of $\mathbb{P}^2$ at one point $p$. Let $D$ be the full preimage of a line in $\mathbb{P}^2$ passing through the point $p$, consisting of the exceptional divisor $E$ and the proper transform $H-E$ of the line through $p$. The Zariski decomposition of $D$ is $D = (H-E) + E$, and $D\cdot E=0$.