For a number of years, Burrows was involved with film director Mike Figgis, and starred in some of his films, including Miss Julie. She was involved with bisexual actor Alan Cumming for several years, while Cumming was separated from his wife. She was romantically linked with Irish actress Fiona Shaw. 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.
In 2006, the Independent on Sunday listed Burrows as the 90th most influential gay person in the UK. In a 2003 interview, actor Alan Cumming said, "I was really lucky that the first relationship I had after [my divorce] was with Saffron, who's really... understanding and a broadminded person. And who's now, as I'm sure you know ... well, she bats for both teams too."
Linked by William Teach at Pirate's Cove in his weekly "Sorta Blogless Sunday and Pinup", with links and Rule 5 posts. Over at The Other McCain, Wombat-Socho has posted the weekly master list "Rule 5 Sunday: Quiet Splendor."