Hello, Pablo. That's a good question. When something is 'wearing thin', it is diminishing. For example, it is common to say that one's patience is wearing thin. My guess is that by 'love line' they mean that their feelings (of love) are diminishing or beginning to lessen. If you are interested in increasing your English fluency, I am a professional teacher on Italki. I'm currently accepting new students and would be happy to work with you.

My love line is wearing thin means my feelings for my the person I love are diminishing.

I took that phrase from my favorite song https://www.youtube.com/watch?v=OOjGMXLvcM4

It's unclear to me, could you give more context? Probably something about his love becoming fragile, but this is not a common expression.