No, but it's also not that simple.
While you do not need to get permission from the songwriters directly to play cover of their songs live, the venues that you are playing at need to obtain the proper licenses with the relevant PROs.
The Copyright Act states that the owner or operator of an establishment or business where music is played is responsible for the performances on the premises and in turn are responsible for obtaining the required authorization(s).
Similarly, the compensation that you receive will not relieve them of their licensing obligation either.
When you play live shows, you are entitled to royalties for your performances separate from the compensation you receive from the venue. However, in order for you to receive royalties, the venue must have the proper licenses. Make sure that you report any covers that you perform on your set lists as well so that the songwriters of those songs can be paid royalties for your performance!
You can learn more about set lists and how to submit them in this section of our help center.