Burch was born in Diamond Bar, California and attended Yale University where she earned a bachelor's degree majoring in Theatre Studies.

In 2014 she moved to London, where she achieved wider acclaim in television following her win in the annual Funny Women competition in the following year,[6] and Fringe First Award for her self written, one-woman show, Tar Baby.

[7] Burch has appeared on Live at the Apollo,[8] Have I Got News for You,[9] QI,[10] The Mash Report,[11] The Russell Howard Hour,[12] The Fake News Show,[13] Chris Ramsey's Stand Up Central,[14] Frankie Boyle's New World Order,[15] Mock the Week,[16] American Autopsy,[17] Richard Osman's House of Games[18] and Would I Lie to You?.

Burch hosts Flinch, a physical game show on Netflix,[22] alongside Seann Walsh, and Lloyd Griffith,[23] which debuted in May 2019.

[26] In May 2020 Burch appeared as a contestant in BBC’s I'll Get This, in which participants play games during dinner to avoid paying the bill.