If I remember correctly, over any field of characteristic $2$, there is a standard example of a smooth conic $\mathcal{C} \subset \mathbb{P}^2$ and a point $a \in \mathbb{P}^2$ such that all lines going through $a$ are tangent to $\mathcal{C}$.

Hence, the projective dual to $\mathcal{C}$ is reducible : it contains a line (namely $a^{\perp}$ and something else). So that $(\mathcal{C}^*)^* \neq \mathcal{C}$.

If I remember correctly, over any field of characteristic $2$, there is a standard example of a smooth conic $\mathcal{C} \subset \mathbb{P}^2$ and a point $a \in \mathbb{P}^2$ such that all lines going through $a$ are tangent to $\mathcal{C}$.

Hence, the projective dual to $\mathcal{C}$ is not integral. In the comments below, Noam notices that $\mathcal{C}^*$ is $a^{\perp}$ with multiplicity $2$. Hence the dual of $\mathcal{C}^*$ is not even well-defined.