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.

If you are including games as part of “math”, chess provides some nice unsolved problems due to computational limits. The game of chess itself cannot even be weakly solved (https://en.m.wikipedia.org/wiki/Solved_game#Overview). But strong solutions are known for a subset of chess positions, those with seven or fewer (total) pieces on the board. These are called (endgame) tablebases: https://en.m.wikipedia.org/wiki/Endgame_tablebase#Background. Any position with eight or more pieces is currently at or beyond present computational resources (chess games start with 32 pieces).

Another source of difficult computation around chess is counting total positions (of certain types) after a certain number of moves. Such as the number of chess games ending in checkmate in exactly N plies (moves by one side), which is presently only known for N <= 13: https://oeis.org/A079485. Or just the total number of possible chess games consisting of N plies, which is presently only known for N <= 14: https://oeis.org/A048987.