Do I need to obtain the rights to a cover song for a live stream?

My band is supposed to perform a setlist of songs at a venue/studio that sells subscriptions to the live stream of our performance on their website. After we shared our setlist with this venue they informed us that we can't perform any covers because they don't have the rights, and asked us if we do.

Which rights or licenses would we need? Since it's a cover song we wouldn't be using the master recording, we'd be playing live. Also, do I/my band need to obtain a license or is that something the venue/studio must do?

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Your help and advice are appreciated.