Two excellent examples of programming with non-constructive proofs are given in Simplifying Programs Extracted from Classical Proofs by Yevgeniy Makarov. The examples they give are better than the immediate example I think of: the proof that there are some two irrational numbers x and y such that xy is rational. This proof is simple to do, but requires some heavy background machinery for rational and irrational numbers. Perhaps I could code this proof by using _|_ for the theorems about rational numbers that I know to be true, but don't wish to prove.
The other two examples of simple non-constructive proofs that I think of are the irrationality of the square root of 2 and the infinitude of the primes. Of course, those are usually stated in non-constructive ways, but they are actually constructive with only a slight bit of tweaking. I haven't read enough of the Makarov paper to see if the simplifications proposed would turn these non-constructive proofs into constructive ones.