This might be a strange question...
Proving there are infinitely many primes numbers can be done by contradiction where we suppose there are finitely many primes and show a contradiction.
However, is there a proof/theorem out there where we suppose a certain set of numbers is infinite and show a contradiction to prove it is finite? Clearly, there are certain sets that are finite that fall under this category. However, I was wanting to know if there was a set that really, truly seems like it could be infinite but it turns out not by a proof by contradiction.
I don't think I know of any "natural" result showing that some specific set can only be proved infinite via a contradiction. However:
There are several results involving fast-growing functions that perhaps are of this kind. Typically, one describes a certain recursive procedure that, when iterated starting from a positive integer, seems to produce larger and larger numbers, and yet the result is that the procedure always "stabilizes" or converges. The proofs tend to involve a detour through infinite ordinals. The argument by contradiction is that if the set of values resulting from the process is truly infinite, the associated ordinals form an infinite decreasing sequence.
Stephen Simpson has a very pretty paper that serves really well as an introduction to these results.
It should be possible to find the paper online, or see here.
An example different from the ones Simpson discusses in some detail is the Hercules-Hydra game (but note that several papers in that volume deal with versions of this game). A "hydra" in this context is represented by a finite rooted tree. In the game, Hercules chops off bits of the tree (heads), and as a result new ones come out, according to certain rules, with more and more heads being generated as the game progresses. Very quickly it seems that even very innocent-looking hydras lead to rather wild ones, and it seems natural to expect that the process continues like this forever, no matter how Hercules proceeds. The surprising fact is that this is not the case, and Hercules always wins after finitely many steps, no matter how he plays. A proof can be achieved by associating to each tree an ordinal, expressed in Cantor normal form (that is, as a sum of ordinal powers of $\omega$). This can be done in a way that, even though the trees appear to grow for a while, the ordinal associated to a tree after a move by Hercules is always smaller than the ordinal associated to the original tree. The sequence of hydras that show up in a game thus leads to a decreasing sequence of ordinals, which should therefore be finite. You may enjoy playing a version of the game, in this page.