Winks, 21, was taken off on a stretcher in the first half of his side's 2-0 win after falling into the dugout.
He made his debut in November 2014, has made 33 appearances this season and scored one goal.
Spurs currently have Danny Rose, Erik Lamela and Harry Kane on the sidelines before Wednesday's trip to Swansea.
However, £30m summer signing Moussa Sissoko, who replaced Winks at Turf Moor, is fit to fill in.
England Under-21 international Winks signed a new contract to extend his terms at White Hart Lane through to 2022 in February.
Tottenham midfielder Harry Winks has been ruled out for the rest of the season by an ankle ligament injury sustained against Burnley on Saturday.