Fire consuming nothing at first, except this eatery (5,3)
I believe the answer is:
snack bar
'this eatery' is the definition.
(type of eatery)
'fire consuming nothing at first except' is the wordplay.
'fire' becomes 'sack' (dismiss someone from their job).
'consuming' indicates putting letters inside.
'at first' suggests taking the first letters.
'except' becomes 'bar' ('bar' can mean 'except' or 'not including').
The initial letter of 'nothing' is 'n'.
'sack' enclosing 'n' is 'snack'.
'snack'+'bar'='SNACK-BAR'
(Other definitions for snack bar that I've seen before include "Lunch counter" , "Lyons, once?" , "Where light meals may be bought" , "Place serving light meals" , "Place to obtain light refreshments" .)