Face extremely greasy — it's humid (5)
I believe the answer is:
muggy
'humid' is the definition.
(muggy can mean humid or close)
'face extremely greasy' is the wordplay.
'face' becomes 'mug' (informal term for face).
'extremely' suggests removing the centre (only the extremes of the word are used).
'greasy' with its middle taken out is 'gy'.
'mug'+'gy'='MUGGY'
'it's' is the link.
(Other definitions for muggy that I've seen before include "Oppressively humid" , "Warm and wet" , "Warm and humid" , "Damp and stifling" , "close" .)