- Diagnosed with dyslexia as a child and struggled academically as a result; his parents eventually enrolled him in a Steiner School.
- Participated with the National Youth Theatre and the Young Arden Theatre as a child.
- Played Romeo in a 2004 production of Romeo and Juliet staged at Shakespeare's Globe Theatre in London.
- Appeared in Creditors at the Donmar Warehouse in 2008; the play was directed by Burke's godfather, actor Alan Rickman.
- Played Greg in Neil LaBute's play reasons to be pretty, opposite Billie Piper, in the 2011 Almeida Theatre production.
- A patron of the Box Clever Theatre, a company dedicated to establishing contemporary theatre groups for young people.