What are good/interesting examples of theorems than can be proven classically, but not constructively, and have applications in e.g. physics?



Applications of nonconstructive mathematics
In general it is very difficult to be sure that a theorem cannot be constructivised in some form that preserves its applicability. As you will notice most of the answers offered have comments attesting this fact.


One reason for this is that many non-constructive theorems in analysis become constructive when they are relaxed a little bit. A typical example is the mean value theorem. It does not hold constructively as usually stated, but it its $\epsilon$ version does: if $f$ is continuous and $f(0) < 0 < f(1)$ then for every $\epsilon > 0$ there is $x \in [0,1]$ such that $|f(x)| < \epsilon$. Many other theorems can be relaxed in this way: Hahn-Banach, Brouwer fixed-point, etc. Moreover, such relaxed versions often make *more sense* in applications than their exact versions, for example because we need to take into account noise, errors, or bounded numerical precision.


Another reason is that for applications we typically do not need a theorem in its full generality because we have extra information, which allows for a specialized constructive version. For example, while the general mean value theorem fails constructively, it holds for locally non-constant maps. Special versions of Hahn-Banach holds constructively, and they will typically suffice in concrete situations.


There is a third reason why it is difficult to find classical theorems with applications that cannot be constructivized. Most applications belong to the fields of physics and computer science, which are both very naturally constructive. Physics is constructive by its very nature because "everything is continuous" in the real world, while in computer science "everything is computable". These are two main motivations for intuitionistic mathematics (namely Brouwerian continuity principles or sheaf-theoretic models, and computable interpretations of intuitionistic mathematics).


Lastly, there remains the simple fact that one has to perform an exhaustive literature search to be sure that a theorem has not been constructivized. A lot more has been constructivized than one would think, and the only obstacle seems to be lack of man power.


