My partner is asexual. Should we break up?
Sadly, there're no easy answers to this. Almost by definition, you're in a situation where there's a tension between you and your partner. Breaking up is not the only option, and should usually be considered a last resort, but it should always be on the table. If your need to sex is very important to you and your partner cannot provide it, then breaking up should be seriously considered.
That said, there are often options for compromise – some asexuals are fine with, or even enjoy, having sex.