The Kenyan international was replaced by Mousa Dembele in the 44th minute after the former Southampton midfielder picked up a knock on his back. Minutes later, Winks slipped off the pitch and injured his ankle after a fall onto Burnley's dugout.
Winks received oxygen on the side and was stretchered off before being rushed to hospital as he was in pain. Mauricio Pochettino confirmed the 21-year-old's ankle injury, while also revealing that Wanyama "suffered a big injury" on his back. The Argentine manager revealed Tottenham will have to wait and see how long the duo will be on the sidelines.
"It's a problem in his [Winks'] ankle. Now, we need to wait for results. We hope it won't be a big issue but we'll see. It's sad, always. Harry is very young and an important player for us and it's always bad news when a player gets an injury," Pochettino explained, as quoted on Tottenham's official website.
On Wanyama, the Tottenham manager said: "Victor suffered a big injury on his back and we'll see now what happens. We're sad for both players and now we have to wait."
Winks and Wanyama join Harry Kane and Erik Lamela at White Hart Lane's treatment room. The England international striker confirmed he suffered ankle ligament damage in his side's FA Cup quarter-final victory over Milwall in March.
Despite suffering injury setbacks against Sean Dyche's side, Tottenham managed to walk away from Turf Moor with three points as Eric Dier and substitute Heung-Min Son were on the scoresheet for the north London club.
Chelsea suffered a 2-1 defeat at the hands of Crystal Palace and their latest defeat has seen them reduce the lead at the top to seven points. Tottenham, who have 62 points after 29 games, are sitting second on the league table.