The two-variable fragment $\mathrm{FO}^2$ consist of those sentences of first-order logic $\mathrm{FO}$ in which precisely two variables occur (e.g. $\exists x \exists y \exists z R(x,y,z)$ is not a sentence of $\mathrm{FO}^2$). In [1] it is proved that the validity problem of $\mathrm{FO}^2$ is coNEXPTIME-complete. What about intuitionistic $\mathrm{FO}^2$? Does it have the same complexity or is it harder? For comparison, the validity problem of propositional logic is coNP-complete while the validity problem of propositional intuitionistic logic is PSPACE-complete, see [2].
This question could of course be asked in more general terms by replacing $\mathrm{FO}^2$ with any reasonable subset of $\mathrm{FO}$. If there is a more general connection between the classical validity problem and the intuitionistic validity problem, then I would be happy to hear about it.
[1] Grädel, E., Kolaitis, P., & Vardi, M. (1997). On the Decision Problem for Two-Variable First-Order Logic. Bulletin of Symbolic Logic, 3(1), 53-69.
[2] Statman, R. (1979). Intuitionistic propositional logic is polynomial-space complete. Theoretical Computer Science, 9(1), 67-72.