Let $\Delta$ be some $n$-simplex. Note that if some $n-1$-dimensional affine hyperplane $U$ intersects $\Delta$, then $U$ separates the vertices of $\Delta$. Therefore, if $T$ is some spanning tree of the vertices then $U$ must intersect $T$. It follows that a sufficient condition for $U$ not intersecting $\Delta$ is $U$ not intersecting $T$. (With spanning tree I mean a union of edges that contains no triangles such that every vertex belongs to one of the edges).
Now suppose that $U$ is $n-k$-dimensional. Then can we find some minimal family $T$ of $k$-faces of $\Delta$ such that $U$ does not intersect $\Delta$ if and only if $U$ does not intersect $T$? I find this a difficult problem.