$C_1 \subseteq C_2$ implies $C_1^A\subseteq C_2^A$?
A proof technique is (informally) relativizing if the results it generates also hold relative to an oracle. Not all proof techniques are relativizing. Perhaps the best known example is $\mathsf{IP}=\mathsf{PSPACE}$, which uses the technique of algebraization. Although $\mathsf{PSPACE} \subseteq \mathsf{IP}$, there is an oracle $O$ such that $\mathsf{PSPACE}^O \not\subseteq \mathsf{IP}^O$ (see this question on cstheory).