'Renege' - break an agreement. The song goes:

Remember when we were hand in hand
Remember we sealed it with a golden band

It's more than just 'giving up' on the relationship; he's accusing her of breaking a promise. (I'm assuming a male singer, here.)

And yes, 'renege on our love' sounds odd, and no, I can't imagine anyone saying it. Perhaps RT wants to emphasise the singer's desparation - and rather posessive attitude to the woman.

I do agree that, in most circumstances, natural words are best - if it expresses what you intend.