Greed's a sin eating inside of tart (7)
I believe the answer is:
avarice
'greed's' is the definition.
(avarice is a kind of greed)
'a sin eating inside of tart' is the wordplay.
'sin' becomes 'vice'.
'eating' means one lot of letters goes inside another.
'inside of' says to take the centre.
The middle of 'tart' is 'ar'.
'vice' enclosing 'ar' is 'varice'.
'a'+'varice'='AVARICE'
(Other definitions for avarice that I've seen before include "Parsimony" , "One of seven deadly sins" , "Greed for gain" , "A deadly sin" , "Greed, covetousness" .)