Why is "I won't forgive you" a commonly used threat?
It's a culture thing, and it's kinda like a fixed phrase, which is not translated accurately into English. If you can understand a little Chinese, its true meaning is 我饶不了你, in which 我(means I) 饶不了(means won't forgive) 你(means you), and the meaning is perfectly translated.

Its true meaning in English is more like this: I will not absolve you from guilt or I will not remit your punishment! (but these are too strong)

For your better understanding, this phrase suggests that if you do anything unpleasant to me, I will remember that one and find a way to make you pay.

PS: Even so it's supposed to be a threat, but this phrase is used between "frienemies" in many cases. For a instance, A doesn't want his partner-like rival B risks his(B's) life to save him(A), then A may say I will never forgive you if you risk your own life to save mine.