No, I don't! This is not standard English. It probably counts as non-standard or even dialect. Standard English would be 'pay any attention' or 'take any notice'. I have only come across this expression in the speech of people from the southern States and the Caribbean. It may be outdated - or 'old-timey' as Benton says - but I would disagree with Jewel about it being formal. I believe it is the kind of phrase you'd be more likely to hear from less educated people these days, and has more of a rural, 'folksy' sound to it.

I have probably heard it less than four times in my entire life in the UK. I was not particularly aware that it was or might be American. Here is a modern Song using it in the lyrics. Undoubtedly an American singer. https://www.youtube.com/watch?v=q-CjjIOlHWg

It's definitely used but not super often. It sounds a bit formal/outdated. "Don't pay them any mind"... I'm young but use some "outdated" phrases myself Haha. But "don't pay attention to them" or "don't worry about them" or "just ignore them" are more colloquial.

As an American living in a very rural area (the mountains of southwest Virginia), I probably hear it used more than Americans living in urban areas, but still not very often. You would more likely hear the sentence "Don't pay any attention to (him/her)."

In American English, I would say that it's used from time to time, but I would say that it's a bit more old-timey. I've said it a few times, but it's definitely not super common in my opinion.