Playwright's quietly leaving Bury (5)
I believe the answer is:
inter
'bury' is the definition.
(to inter is to put into the ground)
'playwright's quietly leaving' is the wordplay.
'playwright' becomes 'pinter' (Harold Pinter).
'quietly' becomes 'p' (p for piano and quiet).
'leaving' is a deletion indicator.
'pinter' with 'p' taken out is 'INTER'.
(Other definitions for inter that I've seen before include "Lay to rest" , "Place in ground" , "Prefix meaning in the middle" , "promoting cessation" , "somewhere between ..." .)