For all oracles A, If $P^A \neq PSPACE^A$, then Does it imply that $P \neq PSPACE$?
If $A = \emptyset$, or more generally, if $A \in \mathsf{P}$, then $\mathsf{P}^A = \mathsf{P}$ and $\mathsf{PSPACE}^A = \mathsf{PSPACE}$, and so $\mathsf{P}^A \neq \mathsf{PSPACE}^A$ is the same as $\mathsf{P} \neq \mathsf{PSPACE}$.
It is also interesting to ask whether $\mathsf{P}^A \neq \mathsf{PSPACE}^A$ for a random oracle implies $\mathsf{P} \neq \mathsf{PSPACE}$. This was disproved (with $\mathsf{P}$ replaced by $\mathsf{IP}$) by Chang, Chor, Goldreich, Hartmanis, Håstad, Ranjan, and Rohatgi in their paper The random oracle hypothesis is false.