Fire getting cold and greyer (7)
I believe the answer is:
cashier
'fire' is the definition.
(I've seen this before)
'cold and greyer' is the wordplay.
'cold' becomes 'c' (eg on taps).
'and' is a charade indicator (letters next to each other).
'greyer' becomes 'ashier' ('ashy' can be similar in meaning to 'grey').
'c'+'ashier'='CASHIER'
'getting' is the link.
(Other definitions for cashier that I've seen before include "I search (anag) - dismiss from armed forces with disgrace" , "Teller - dismiss in disgrace" , "Shop payment taker" , "money dispenser" , "may be a shop-worker" .)