Runnett was killed in a car crash in 1970 whilst returning from an organ recital he had given in Westminster Abbey. Composer Kenneth Leighton was commissioned by the Cathedral Organists' Association to write a memorial work for Runnett - his Second Evening Service (Magnificat and Nunc Dimmitis).
Norwich Cathedral has a Runnett Library in Runnett's memory, which contains all of his sheet music. It was donated to the cathedral by Runnett's parents following his death. St John's College, Cambridge also has a Brian Runnett Memorial organ competition, as well as trophies given out at the end of it..