Unsolved Problems due to Lack of Computational Power

Goldbach's weak conjecture isn't a conjecture anymore, but before it was proved (in 2013), it had already been proved that it was true for every $n>e^{e^{16\,038}}$. It was not computationally possible to test it for all numbers $n\leqslant e^{e^{16\,038}}$ though.

Some notorious problems of this kind are in discrete mathematics but involve a search space that is many magnitudes beyond what is feasible. For example, the values of certain Ramsey numbers or the existence of a Moore graph of degree 57.

Historically, a very important, computationally intensive problem arising from physics was lattice QCD (LQCD). LQCD is a theoretical framework for computing basic quantities like the mass of the proton, and it was introduced by Ken Wilson back in the 70's. However, after some initial successes, this approach stagnated due to a lack of computer power. The basic problem is that our universe has an obnoxiously large number of dimensions (four, in case you were wondering), and doing integrals in four dimensions takes an insane amount of memory. I heard a story that Ken Wilson gave a talk at a conference on LQCD where he declared that "Lattice QCD is dead" as long as a certain 4D integral could not be computed, as was the case at the time he said this.

Several years (or decades) later, computer technology matured to the point that said integral could be computed, and then LQCD theory picked right back up where it left off. Today it is again a flourishing discipline. However, other problems arising from LQCD continue to push supercomputer technology. Apparently LQCD is used as a benchmark for supercomputers nowadays.