No. The royalty rules are different based on the medium (live or recorded), but you never need an artist's permission to cover a song.
In regards to live performance, specifically, the venue is responsible for obtaining a blanket license through a performance rights organization for all copyrighted songs that would be played in their venue. The artist doesn't need to worry about it, and can just show up and play their gig. The music lawyer says:
It is generally the responsibility of the venue owner (i.e., the presenter of the public performance), not the performer, to obtain a public performance license and pay any required licensing fees.
As a practical matter, venue owners obtain blanket licenses from the performing rights organizations (ASCAP, BMI, and SESAC, in the U.S.) to have the right to present musical performances at their venues. That's because it's not feasible for venue owners to research the ownership of each song and negotiate individual public performance licenses. (Imagine the venue owner requiring you to give them your set list weeks in advance and making you promise not to deviate from the list!)
Performance rights organizations not only make life easier for the cover artists, but also the artists who hold the copyright to their songs:
Performing rights organizations (or PROs) also serve the songwriters and publishers who can't be expected to police all the venues in the country to see who is performing their music. Of course, the PROs only pay out to those songwriters and publishers who have affiliated and have registered their songs in the PRO catalog. Therefore, it's important for songwriters and publishers to keep their catalog up-to-date to ensure that they are getting the most public performance monies.