Minimum spanning tree of a connected induced subgraph
Suppose that $G$ is the triangle on $\{1,2,3\}$ (with arbitrary edge weights), that $T$ is $\{\{1,2\},\{1,3\}\}$ (without loss of generality), and consider $H = \{\{2,3\}\}$, which is induced by $S = \{2,3\}$. No edges of $T$ lie in $H$, and in particular these edges do not constitute a spanning tree of $H$.