Playwright quietly leaving Bury (5)
I believe the answer is:
inter
'bury' is the definition.
(to inter is to put into the ground)
'playwright quietly leaving' is the wordplay.
'playwright' becomes 'pinter' (Harold Pinter).
'quietly' becomes 'p' (p for piano and quiet).
'leaving' indicates named letters should be taken away.
'pinter' with 'p' taken away is 'INTER'.
(Other definitions for inter that I've seen before include "promoting cessation" , "Cover" , "somewhere between ..." , "football team abroad" , "Prefix meaning between or among" .)