Yes - Songtrust will collect the publisher's share of the royalties you earn from submitting your setlists. Live performances are a source of Performance Royalties. These royalties are split the same as all other royalty sources (learn more here).
US Societies like ASCAP and BMI require you to submit your setlists to them directly in your PRO account outside of Songtrust, but Songtrust will still collect in those instances regardless.
You can learn how to submit you setlists in this demo!