Rather surprisingly the answer is yes, particularly in the area of computational number theory. In the most famous example, Gary Miller in 1975 gave a polynomial-time algorithm for primality whose correctness could be proven by assuming the Extended Riemann Hypothesis (ERH). Of course in 2002 we had a polynomial-time primality algorithm with no assumption. However the original analysis of the algorithm gave a constant which depends on how ERH is resolved.
There are still many other problems in computational number theory that require ERH. For example, according to Eric Bach, the only polynomial-time algorithm computing square roots modulo p, when p is large relies on ERH. "The idea is to combine an algorithm that uses a quadratic nonresidue, such as Shanks's algorithm (this in Knuth v. 2 I am pretty sure) with a bound on the least quadratic nonresidue mod p (e.g. in my thesis it is proved to be <= 2 (ln p)^2 if ERH is true)."