0310 GMT January 23, 2020
Mauricio Pochettino's side would have moved within two points of leaders Manchester City with a win at Turf Moor, AFP reported.
But it squandered the chance to enhance its title prospects with a sloppy display that saw Chris Wood put Burnley ahead after halftime.
There was controversy about Wood's goal as it came from a corner that Tottenham felt shouldn't have been given.
But Pochettino's men had a break of their own when Kane, back after missing seven games with an ankle injury, capitalized on a throw-in that was taken in the wrong place.
At that stage, Tottenham looked poised to push on for a fifth successive league victory.
However, it paid the price for sloppy defending as Barnes struck with seven minutes left to extend Burnley's unbeaten league run to eight matches.
It was a bitter blow for third placed Tottenham, who will be eight points off top spot if second placed Liverpool, currently level on points with City, win at Manchester United today.
Pochettino was fuming at full-time, confronting referee Mike Dean and his assistants on the pitch in a lengthy rant that showed his frustration at both Burnley's first goal and Tottenham's missed opportunity.
Playing for the first time in 10 days due to its early exit from the FA Cup, Tottenham was back in action after beating Borussia Dortmund in the Champions League in its previous match.
Pochettino knew his team had no margin for error if it wanted to stay in contention and, as games against Chelsea and Arsenal loom next week, he billed the trip to Turf Moor as a "final".
With England manager Gareth Southgate watching from the stands, Kane came close to marking his return in style when he fired narrowly wide from the edge of the area.
But Tottenham was struggling to build the head of steam required to break down the Burnley’s well-marshaled rearguard.
Kane, back to fitness ahead of schedule, grew more menacing as he shook off the rust and early in the second half the striker drew a superb tip-over from Tom Heaton when he let fly from 25 yards.
Moments later, Pochettino was fuming on the touchline as Burnley snatched the lead with Wood's seventh goal in his last 10 games in the 57th minute.
When Tottenham defender Jan Vertonghen slid in to challenge Jeff Hendrick, a corner was given to Burnley even though the ball appeared to come off Hendrick last.
From Ashley Westwood's corner, Tottenham went to sleep as Wood timed his run to climb above the statuesque Juan Foyth and Toby Alderweireld.
The New Zealand striker's powerful header hit the crossbar and bounced over the line before Tottenham goalkeeper Hugo Lloris could react.
Pochettino complained bitterly to the fourth official, but the Argentine's protests were quelled as Tottenham grabbed a controversial goal of its own eight minutes later.
Danny Rose took a Tottenham throw from several yards further forward than the spot the ball went out, and his delivery sent Kane racing clear on goal.
Kane showed he had lost none of his predatory instincts as he held off Ben Mee and slotted a cool finish past Heaton for his 21st goal of the season and his first since early January.
Tottenham had stolen late winners in three of its last four league games, but it was Burnley which finished in the ascendency in the 83rd minute.
Johann Berg Gudmundsson seized possession in midfield and drove into the Tottenham area unchecked by the slow-to-react visitors.
Gudmundsson smashed a cross-shot towards the far post and, with Vertonghen unable to clear, Barnes had the simple task of tapping into the empty net.