Do well to shut up! (7)
I believe the answer is:
confine
'shut up' is the definition.
(thesaurus)
'do well' is the wordplay.
'do' becomes 'con'.
'well' becomes 'fine' ('fine' can be similar in meaning to 'well').
'con'+'fine'='CONFINE'
'to' is the link.
(Other definitions for confine that I've seen before include "Limit" , "Shut in" , "Imprison" , "Incarcerate" , "Keep within limits" .)