Kane limped off in the 10th minute at White Hart Lane after his ankle twisted awkwardly.
The England striker was sidelined with another ankle problem earlier this season and the injury is a major worry for Tottenham boss Mauricio Pochettino.
England manager Gareth Southgate will also be concerned as he is due to name his squad for forthcoming fixtures against Germany and Lithuania.
Tottenham are looking to win the FA Cup for the first time since 1991 and close the gap on Premier League leaders Chelsea, but those ambitions would be much harder to achieve without their leading scorer.
Despite Kane's absence, Tottenham were too strong for third tier Millwall and they marked the last FA Cup tie at the Lane before the stadium is demolished, with a commanding quarter-final display.
Christian Eriksen opened the scoring and Dele Alli and Vincent Janssen were also on target, but it was Son who took centre-stage with his treble.
Tottenham will discover their opponent for next month's Wembley semi-finals on Monday, but news of Kane will be anticipated even more eagerly.
Given the notorious reputation of Millwall's support, much of the pre-match focus had been on the possibility of crowd trouble.
There was a tense atmosphere outside the stadium before kick off, but a huge police presence managed to contain the threat of serious violence despite bottle throwing and minor scuffles.
Inside, a large segregated area was used to keep Millwall and Tottenham fans apart, with a line of stewards wearing orange hard hats in case missiles were thrown.
Millwall arrived with a 17-match unbeaten run that included FA Cup upsets of Bournemouth, Watford and Leicester, but facing a team of Tottenham's class was a different proposition altogether.
Dominating possession with ease, Pochettino's side battered Millwall from start to finish as they cruised to a 16th win from 18 home games this season.
Kane's injury came early, the England star's ankle twisting badly after Jake Cooper slid into him.
After several minutes of treatment, Kane limped off to be replaced by Danish winger Eriksen.
It was a big blow for Tottenham given Kane's red hot form had brought him eight goals in his last four appearances and 24 for the season. As Tottenham regrouped, Millwall striker Steve Morison curled wide from 20 yards in a rare moment of alarm for the hosts.
But Tottenham's incessant pressure was rewarded in the 31st minute when Alli chested down a high clearance and Eriksen pounced to drill a low shot into the far corner from just inside the area.
Victor Wanyama powered a header against the crossbar as Tottenham pressed for the second goal, which arrived in spectacular fashion from Son in the 41st minute.
Picked out by Eriksen on the edge of the area, Son cut inside and curled a sublime shot past King.
Son's first goal since January was the perfect way to silence some unsavoury taunts aimed at him by Millwall fans, and the South Korean wasn't finished yet.
Making a perfectly-timed run onto Kieran Trippier's pass, Son got in behind the Millwall defence in the 54th minute and volleyed a fine finish past King.
Alli tapped in Eriksen's cross for the fourth in the 72nd minute and Janssen made it five seven minutes later with his first goal from open play for Spurs since signing last year.
There was still time for Son to be gifted his treble in stoppage-time when King let his shot slip through his legs into the net.