On $SUBEXP$, $PP$ and $P/poly$
If $D$ is low for itself ($D^D = D$) then $C,O \subseteq D$ should imply $C^O \subseteq D$, though it might depend on the exact definition of the oracle model (for resource-restricted classes there are sometimes delicate issues there). In contrast, if $D$ isn't low for itself then taking $C=O=D$ we get a counterexample.