The first two things that come to my mind are actually the dual of what you asked for, namely conjectures where the counterexample did *not* come first, and the search for an explicit counterexample led to interesting developments.

1. <a href="https://arxiv.org/abs/math/0012198">Brosnan and Belkale</a> disproved a conjecture of Kontsevich about polynomially countable graphs, but their argument did not lead to a specific counterexample.  An explicit counterexample was provided by <a href="https://arxiv.org/abs/1006.3533">Dzmitry Doryn</a>.

2. That $\pi(x) > \mathrm{li}(x)$ for all $x$ was disproved by Littlewood, but without giving an explicit counterexample.  I believe that there is still no explicit counterexample, but the search for one has inspired some <a href="https://en.wikipedia.org/wiki/Skewes%27s_number">interesting work</a>.