Burns, 26, was named as players' player of the year last season and scored over 1,000 Championship runs.
Foakes, 23, averaged over 40 with the bat and will tour during the first part of England Lions' winter programme.
Curran, younger brother of Tom, has already played 16 first-class matches for Surrey at the age of 18.
He will also be part of the Lions squad this winter.
Vice-captain Rory Burns, wicketkeeper Ben Foakes and all-rounder Sam Curran have signed new three-year contracts with Surrey.