Burrows is a close friend of politician Tony Benn. She is also a socialist and has spoken of her admiration for French Socialist politician Ségolène Royal. She joined an anti-racism group when she was 11 years old and she went on to become the Vice President of the National Civil Rights Movement.

In a 1999 interview with Film Unlimited, Burrows revealed that she is bisexual. For a number of years, she was involved with film director Mike Figgis, and she starred in some of his films, including Miss Julie - one of the first films to experiment with multi-camera views, and Figgis' quad-screen film, Timecode.

Since the ending of that relationship, she has been romantically linked with actress Fiona Shaw. Neither actress has publicly confirmed or denied the relationship. The two appeared together in the National Theatre's production of The PowerBook, a play based on the novel of the same name by Jeanette Winterson, in which they played lovers.