Religious martyrs with no political agenda who go to certain death rather than deny their religion are literally "dying for a belief."

In contrast, political martyrs like the IRA hunger strikers who starved themselves to death die as much in the hope of improving conditions for others as they do simply to uphold an abstract principle. That's a belief, too, in the sense that there's no proof that it's going to happen, but the nature of the hoped-for result is quite different.

How "sane" such martyrs are, or whether overall "sanity" is even relevant at such times, is an interesting question. In some cases, it might just be irrational defiance.

It's a very good thing that more people aren't ready simply to "die for a belief." If they were, many more arguments would end in death, and every war would be a lot bloodier.