Nine days before the Carabao Cup final and 58 days before England’s European Championship opener, Kane limped off in stoppage time with an apparent ankle problem.
The striker seemed to land awkwardly after jumping for a header when defending a corner late on.
Manager Jose Mourinho said he did not know how serious the injury was.
“That’s also a question I want an answer to as soon as possible but it’s too early,” Mourinho said.
“For him to come out it is not because it’s nothing, it’s something.
“We all felt the same fears of something that stops him playing Wednesday [against Southampton in the Premier League] and especially Sunday [the Carabao Cup final] but we will see.”
Kane had earlier been at his clinical best, capitalizing on two mistakes in the Everton defence to put Spurs ahead in the first half and grab a second-half equalizer.
In between Kane’s strikes Gylfi Sigurdsson levelled from the penalty spot and gave Everton the lead by finishing off a fine team goal.
Everton pushed hardest for a winner and had the best chance to win it but Joshua King’s effort was saved by Hugo Lloris and Richarlison fired the rebound over from 12 yards.
A draw suited neither side’s pursuit of European football, with Tottenham staying seventh, five points behind fourth-placed West Ham and Everton a point and a place worse off, though they do have a game in hand.