MathOverflow is a valuable resource because it is interdisciplinary. Some of the people who have written about this question come from a traditional pure mathematical background where classical logic is the norm, whilst others are based in computer science departments and generally use intuitionistic logic(s). Something that even those of us in the latter category perhaps do not credit enough is that some quite substantial "heterodox" mathematics has been done in the past thirty years. This means, for example, that Dan Pipponi's answer presupposes some quite sophisticated ideas.
I suspect that the thinking behind this question is that it is sometimes said that classical mathematicians ought to be willing to acknowledge intuitionistic mathematics in the same way that they do non-commutative group or ring theory. I agree with this professionally, but I do not think that there is a significant mathematical analogy to be made.
If we are looking for analogies in ring theory to the classical/intuitionistic distinction, I think a better one would be the passage from $\mathbb Z$ to (commutative) integral domains in which ideals need not be principal. The development needs to be rewritten, but for the most part this is a matter of "cleaning things up" rather than doing a completely different and vastly more difficult thing like non-commutative ring theory. Unfortunately, rather a lot of the literature is made "dirty" with Excluded Middle, so there is a Herculean task to clean it up.
Treating logic in terms of (Boolean or Heyting) algebra is in most circumstances misleading. At primary school we evaluated arithmetic expressions from the inside out, but then we learned to manipulate ones with indeterminate values. Similarly, logic is not about things that "are either true or false" but which instead may perhaps be deducible from one another. Here I am simply making an observation about what mathematicians actually do, even classical ones.
The deduction operation that is at issue is Excluded Middle.
Essentially, Excluded Middle is like the fear of water. If your parents take you swimming as a baby, maybe before you can walk, then you do not develop the fear of water and learn to swim entirely naturally.
Similarly, if your teachers do not constantly indoctrinate you by beginning every proof in their lectures with "suppose not" then you will naturally grow up to be a constructive mathematician. It is only difficult because you have been told to think it is.
It is common to see arguments that use contradiction quite gratuitously. They are much more complicated because, instead of proving $C\Rightarrow D\Rightarrow E$, they prove $\lnot E\Rightarrow\lnot D\Rightarrow\lnot C$, so the argument is back-to-front. When some parts of a proof that is naturally $$ A\Rightarrow B\Rightarrow C\Rightarrow D\Rightarrow E\Rightarrow F\Rightarrow G$$ are written forwards and others backwards, it turns into spaghetti: $$ A\Rightarrow B\Rightarrow C,\qquad \lnot E\Rightarrow\lnot D\Rightarrow\lnot C\qquad E\Rightarrow F\Rightarrow G. $$
In fact, as Dan has said, there is a lot of work in theoretical computer science based on the idea that the double negation rule is like a "computational effect" (such as exceptions and gotos) in programming. Unless used vary skillfully, such effects make programs next to impossible to understand.
On the other hand, there is considerable skill (that classical mathematicians refuse to acknowledge) in pulling a classical proof apart, teasing out its underlying concepts and creating a new constructive proof.
I would, for example, strongly recommend Constructive Analysis by Errett Bishop and Douglas Bridges, which gets on with proving the theorems without dwelling on the counterexamples.
The reason why some of us regard intuitionistic logic as fundamental and classical logic as an aberration lies in the following analogy (often called an "isomorphism") that was made by Haskell Curry in the 1930s and spelt out by William Howard in the 1960s. This analogy nowadays probably forms the basis of the masters' logic course in any computer science department.
A proof of the conjunction $P\land Q$ is a pair $(p,q)$, where $p$ is a proof of $P$ and $q$ is a proof of $Q$.
A proof of the implication $P\Rightarrow Q$ is a function $f$, where $f(p)$ is a proof of $Q$ whenever $p$ is a proof of $P$.
A proof of the univeral quantification $\forall x:X.P(x)$ is a function $f$, where $f(a)$ is a proof of $P(a)$ whenever $a$ is an element of $X$.
Whilst this might perhaps be seen as begging the question, it is difficult to see how one treat the other two connectives otherwise than:
A proof of the disjunction $P\lor Q$ is a pair $(i,r)$, where either $i=0$ and $r$ is a proof of $P$ or $i=1$ and $r$ is a proof of $Q$.
A proof of the existential quantification $\exists x:X.P(x)$ is pair $(a,p)$, where $a$ is an element of $X$ and $p$ is a proof of $P(a)$.
The point here is that there is no obvious way of translating excluded middle.
I say "obvious" because the work to which Dan refers seeks to do exactly that.
However, it is important to stress that those who are working on this kind of thing have not "seen the error of their ways" and returned to the "true faith" of classical logic, but are doing something that is way more sophisticated.
Returning to the original question, I am very skeptical. But the reason for this is not a lack of faith in intuitionistic logic or to disparage the work that people are doing in quantum mechanics. It is because those who are doing work like this probably use intuitionistic logic as a matter of course and would never consider naive classical logic as an alternative.