This does not seem to be true: take $X_t$ to be a symmetric stable process in $\mathbf{R}$ with index $\alpha > 1$, and $D = (-1, 1) \setminus \{0\}$. Then $X_t$ hits $0$ with positive probability.

Things get even worse when one thinks about non-symmetric processes: if $X_t$ is strictly stable with one-sided jumps and with index $\alpha > 1$, and $D = (-1, 1)$, then $X_t$ hits the boundary with positive probability, even though $D$ is extremely regular.

For one-dimensional Lévy processes exiting an interval this is precisely the question whether the process creeps or not. See, for example, Kyprianou's book *Introductory Lectures on Fluctuations of Lévy Processes with Applications* for further information.

In higher dimensions, if $D$ is regular enough, the answer should be similar. However, I doubt this is rigorously proved anywhere.