Dier returned as a substitute in Sunday's 2-1 win at Fulham after spending over a month out following surgery to remove his appendix.
But striker Kane is now out until March with an ankle injury and Alli suffered a hamstring problem which requires a scan.
The full extent of Alli's injury will be assessed by a scan, the results of which are likely to come in the next few days.
Initial suggestions pointed to it being unlikely the England midfielder will be involved in Thursday's Carabao Cup semi-final second leg at Chelsea.
"Every team gets injuries; we have to deal with it, we're not sulking about it or feeling sorry for ourselves," Dier said.
"It's not a question of if we can cope without Harry or not. Harry's one of the best strikers in the world, any team would feel the loss if Harry wasn't playing, so we have to find a solution and move on.
"Injuries happen in football and that's why you have a squad and not just 11 players. It was a good start (to life without him). People are going to be talking about it when he's injured but we've just got to get on with it. 'Harry will be doing his best to get back as soon as possible and it's down to the team and the squad to rally and carry on.
"Fernando (Llorente) is a very good player. He has very good qualities. We need to give him the service he needs to demonstrate his qualities. We can do that and he can (also) help the team."