Celebrity vet has removed tail fast (6)
I believe the answer is:
starve
'fast' is the definition.
(to fast is to refrain from eating food)
'celebrity vet has removed tail' is the wordplay.
'celebrity' becomes 'star' (a star is a celebrity or famous person).
'has removed tail' means to remove the last letter.
'vet' with its final letter taken away is 've'.
'star'+'ve'='STARVE'
(Other definitions for starve that I've seen before include "Averts (anag)" , "Suffer from food deprivation" , "Suffer extreme hunger" , "Eat nothing" , "Suffer from lack of food" .)