From personal experience, I would say as long as you give the venue your song list identifying the owner of each song (or arrangement) then you are OK. There is a standard form that the venues should have. It is the venue that is responsible for paying for a PRS licence and submitting performance lists. There is no cost to you.

I have requested permission from several owners to perform their material, the response without exception has been total silence. I conclude that you don't need permission to perform someone else's material. It's up to the writer of a song to register it with PRS, once they do that then they will get a royalty payment for each time it is performed.

Copyright is a bit of a different issue, it's clear that you must not copy the printed version of a song for instance without permission (and making a payment perhaps). Whether restrictions apply to performing I'm not so clear on.