Let $(L_t)_{t \geq 0}$ be a $k$-dimensional Lévy process and $σ: \mathbb{R}^d \to \mathbb{R}^{d \times k}$ a continuous function such that the Lévy-driven stochastic differential equation (SDE) $$dX_t = σ(X_{t-}) \, dL_t, \qquad X_0 \sim μ$$ has a unique weak solution. We show that the solution is a Feller process whose domain of the generator contains the smooth functions with compact support if, and only if, the Lévy measure $ν$ of the driving Lévy process $(L_t)_{t \geq 0}$ satisfies $$ν(\{y \in \mathbb{R}^k; |σ(x)y+x|<r\}) \xrightarrow[]{|x| \to \infty} 0.$$