Top-scorer Kane is out until March with an ankle injury and fellow forward Son Heung-min is on duty with South Korea at the Asian Cup.
Alli came off towards the end of Sunday's last-gasp 2-1 win at Fulham with a hamstring problem, giving boss Mauricio Pochettino a headache as he juggles scant resources.
The full extent of Alli's injury will be assessed using a scan, the results of which are likely to come in the next few days but it is unlikely the England midfielder will be involved in Thursday's League Cup semi-final second leg at Chelsea.
Tottenham lead 1-0 after the first leg.
"Every team gets injuries. We have to deal with it -- we're not sulking about it or feeling sorry for ourselves," said Dier, who returned as a substitute at Fulham after spending more than a month out following surgery to remove his appendix.
In the absence of Kane and Son, Fernando Llorente played up front at Craven Cottage.
Fulham took the lead in the first half courtesy of a Llorente own goal but Alli equalised and Harry Winks grabbed the winner deep into injury time.
"It's not a question of if we can cope without Harry or not," said Dier.
"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," added the England international.
Spurs are third in the Premier League and play Borussia Dortmund in the Champions League last 16 next month.