In fault-tolerant distance labeling we wish to assign short labels to the vertices of a graph $G$ such that from the labels of any three vertices $u,v,f$ we can infer the $u$-to-$v$ distance in the graph $G\setminus \{f\}$. We show that any directed weighted planar graph (and in fact any graph in a graph family with $O(\sqrt{n})$-size separators, such as minor-free graphs) admits fault-tolerant distance labels of size $O(n^{2/3})$. We extend these labels in a way that allows us to also count the number of shortest paths, and provide additional upper and lower bounds for labels and oracles for counting shortest paths.