Learn English – A word for forcing oneself to do something one dislikes

Staying within your religious context, this practice is known as

Mortification of the flesh is an act by which a individual or group seeks to mortify, or put to death, their sinful nature, as a part of the process of sanctification.
In Christianity, common forms of mortification that are practiced to this day include fasting, abstinence, as well as pious kneeling. _Wikipedia

Another example of mortification of the flesh is self-flagellation.

A related, and complementary, concept is asceticism, which, as opposed to doing something one finds distasteful, is forgoing or abstaining from things one would like to do.

Asceticism is a lifestyle characterized by abstinence from worldly pleasures, often for the purpose of pursuing spiritual goals._Wikipedia

An alternative to asceticism, suggested by @Cerebus in the comments, is ascesis, which I like better than asceticism because it seems more personal, more of a practice specific to that one individual.

Finally, a common conversational rebuke in these contexts is:

Oh, don't be such a martyr!

Which @Matt S cleverly reduced to martyrdom.