No, intuitionistic logic is not the quantization of classical logic in a useful sense. (It is obtained from classical logic by removing artifical constraints, which is a step rather different from quantization.)
But there is a way to see the point of intuitionistic logic from a perspective motivated from quantization.
To see this, the main point to notice is that intuitionistic logic is the logic that allows to carry out reasoning in ambient toposes other than that of plain sets, hence in geometric contexts. That's the whole point of it.
And this is something that all of quantization theory starts with (usually secretly, of course): the "classical" (really: pre-quantum) data that quantization is applied to lives in generalized differential geometry where path spaces and spaces of differential forms etc. exist as smooth spaces. And this means that all of quantization starts (usually secretly, of course) in an ambient intuitionistic context. Of course there are plenty of ways to fight noticing this, which is why it's still an exotic perspective sociologically, even though it is the natural perspective fundamentally. The lecture notes geometry of physics try to give the natural perspective.
In
- Urs Schreiber, Mike Shulman, Quantum gauge field theory in Cohesive homotopy type theory
an intuitionistic axiomatization of (pre-)quantum physics is laid out, and in
- Domenico Fiorenza, Chris Rogers, Urs Schreiber, Higher geometric prequantum theory
it is is discussed how Heisenberg groups, Poisson Lie algebras, Heisenberg Lie algebras, (pre-)quantum operators etc. drop out naturally from these intuitionistic axioms.
Indeed, once axiomatized in intuitionistic logic this way, all these concept become much simpler than they are usually thought of. They are really sitting just a tad above the very foundations (univalent foundations, that is). See the above articles for why and how.
[edit Jan 2014:
Meanwhile with Joost Nuiten we have further developed the formalization of quantization in intuitionistic linear logic, details are here
- Urs Schreiber, Homotopy-type semantics for quantization
Indeed, if anything in logic resembles quantization, then it is the step from intuitionistic non-linear to intuitionistic linear logic, this is discussed a bit more on the $n$Lab at quantum logic . ]