He is best known for starring as Titus Andromedon on the Netflix comedy series Unbreakable Kimmy Schmidt (2015–2020), for which he received five Primetime Emmy Award nominations.

[3] Born and raised in Athens, Georgia, Burgess attended Cedar Shoals High School where he was active in the theatre program.

[9] Burgess made his Broadway debut in the musical Good Vibrations as Eddie in 2005, and then appeared in Jersey Boys in 2005 as Hal Miller.

[10] Burgess originated the role of Sebastian the Crab in the Broadway musical The Little Mermaid in 2007 opposite Sierra Boggess, Sean Palmer, and Norm Lewis.

[citation needed] He was cast as The Witch, a role traditionally played by a female, in Into the Woods in a 2015 production by DreamCatcher Theatre, presented in Miami's Carnival Studio Theater at the Adrienne Arsht Center.

[citation needed] In June 2016, Burgess and Norm Lewis reprised their roles of Sebastian the Crab and King Triton in the Hollywood Bowl concert event of The Little Mermaid opposite Sara Bareilles, Rebel Wilson, and Darren Criss.

From 2020 to 2022, he starred in the musical cartoon series Central Park as Cole Tillerman, with Leslie Odom Jr., Kristen Bell, Daveed Diggs, Josh Gad, Kathryn Hahn and Stanley Tucci.

[20] Burgess starred as Remy in a benefit concert presentation of Ratatouille the Musical, an internet meme that originated on TikTok, inspired by the 2007 Disney/Pixar film.