I'm not sure whether there is an efficient way to answer your question (or a written reference), but it's possible to analyze the situation case-by-case.

It's probably best to start with an *irreducible* root system $\Phi$ (reduced in the Bourbaki sense), which can be asociated with a simple Lie algebra over $\mathbb{C}$ but can just as well be studied abstractly.  Fix a simple system $\Delta$ and the corresponding Weyl group $W$ generated by the simple reflections $s_\alpha$ (with $\alpha \in \Delta$).     Now choose a (proper) subset $\Delta' \subset \Delta$ with a corresponding root subsystem $\Phi'$ (not necessarily irreducible) and Weyl subgroup $W'$.

It's important to keep in mind that the automorphism group Aut($\Phi$) is the semidirect product of the normal subgroup $W$ and the possibly trivial group $\Gamma$ of graph automorphisms.  An example is given by Nathan Reading for type $A_3$, where the nontrivial graph automorphism has order 2.   Similarly, Aut($\Phi'$) is the product of such automorphism groups, which might or might not be realized within $W$ and might or might not involve graph automorphisms.   So a certain amount of case-by-case description could be needed.

Given this set-up, suppose $1 \neq w \in W$ stabilizes $\Delta'$.    Here $w$ might be a product of simple reflections corresponding to vertices of the Dynkin diagram not connected to any root in $\Delta'$ (so the roots in question are orthogonal).   Or a reduced expression for $w$ might involve a mixture of simple reflections coming from roots in $\Delta'$ and roots in $\Delta \backslash \Delta'$, as in Nathan's example.