Mauricio Pochettino's men find themselves seven points adrift of fourth-placed Manchester City after an uninspired showing against organised and spirited opponents
Kane was handed the armband in the absence of the injured Hugo Lloris, but the England striker was ineffective in a lacklustre display from the London club which ensured they were unable to move above Liverpool into fifth place.
A point was enough for Spurs to leapfrog Southampton into sixth spot, but their chances of securing Champions League qualification look increasingly slim with seven matches remaining.
Burnley deserved at least a point at Turf Moor, but may see this as a missed opportunity a day after QPR and Leicester City secured much-needed victories in the battle to avoid the drop.
Sean Dyche's men have now won just one of their last 10 top-flight games, although a draw enabled them to move back above QPR into 18th spot.
Burnley boss Dyche named an unchanged starting line-up and included recent signing Fredrik Ulvestad on the bench for the first time.
Michel Vorm made his maiden Premier League start for Tottenham due to Lloris' leg injury, while Vlad Chiriches replaced the unwell Jan Vertonghen and Paulinho was preferred to Andros Townsend.
Burnley pressed Spurs high up the pitch from the off and should have been rewarded with a goal after just four minutes, when George Boyd robbed Ryan Mason but Danny Ings shot straight at Vorm after surging into the penalty area.
Nacer Chadli had Tottenham's first sight of goal after 15 minutes, forcing Tom Heaton to get down well to smother a shot from the Belgium winger that came through a crowded penalty area.
Boyd's stunning strike gave Burnley a famous win over Manchester City in their last home game, but the industrious midfielder failed to trouble Vorm when the ball fell kindly for him just outside the area.
Vorm's agility was put to the test just before half-time, though, when he dived to his left to palm over Ings' strike from the edge of the area.
Kyle Walker's afternoon was cut short by an injury just after half-time, so Chiriches moved to right-back and Ben Davies came on at left-back for Tottenham.
Ings was proving to be a handful for the Spurs defence and Vorm had to be alert to palm the striker's attempted cross behind for a corner as Burnley started to gain the upper hand.
Vorm then pushed away Ashley Barnes' strike in a good spell of pressure from the home side, before Christian Eriksen had a good chance to open the scoring against the run of play when he raced into the area and fired over the crossbar.
Boyd headed just wide in stoppage time, but neither side did enough to secure all three points from a forgettable encounter.