If your heart wears thin I will hold you up. And I will hide you when it gets too much.
This is from a song. I couldn’t really find a definition of heart wears thin.
The idiomatic expression is wear thin:
- Be weakened or diminished gradually, as in My patience is wearing thin. [Late 1800s].
- Become less convincing, acceptable, or popular, as in His excuses are wearing thin. [First half of 1990s]
Both usages transfer the thinning of a physical object, such as cloth, to nonmaterial characteristics.
In your sentence heart means courgage, and wearing thin means “lose heart.”