Bury chest, dismissing church's concern (8)
I believe the answer is:
interest
'concern' is the definition.
(synonyms)
'bury chest dismissing church's' is the wordplay.
'bury' becomes 'inter' (synonyms).
'dismissing' indicates named letters should be taken away.
'church' becomes 'ch' (common abbreviation).
'chest' with 'ch' taken away is 'est'.
'inter'+'est'='INTEREST'
(Other definitions for interest that I've seen before include "point of saving?" , "Deep concern - premium paid for use of money" , "Concern or pastime" , "Significance" , "Concern - money accruing" .)