Harry Kane was named captain for Tottenham but had little effect in a scrappy game as Danny Ings had the best chance in the first half, but he struck straight at Michel Vorm after being put through one-on-one by George Boyd.
The draw moves Burnley above Queens Park Rangers into 18th place, two points behind Aston Villa.
Fresh from winning his first caps for England, Kane was given the armband by Mauricio Pochettino in the absence of Hugo Lloris but it was another English striker, Ings, who looked in the mood early on.
With just five minutes on the clock he raced through on goal after Boyd had done brilliantly to dispossess Ryan Mason deep in Tottenham territory.
Nabil Bentaleb flirted with bringing down the striker as he raced towards goal but Ings stayed on his feet to get his shot away. The effort failed to seriously test Vorm though as his strike was too close to the former Swansea goalkeeper.
The game had all the hallmarks of an end-to-end classic at that point, but it failed to hit those heights as the two sets of defences ruled all.
Ings did create a chance for himself on 45 minutes when he worked space 25 yards from goal, but his curling effort was tipped over the crossbar by Vorm.
Kane had just 22 touches in the opening 45 minutes – the least of any Tottenham player, which summed up the struggles Spurs had getting the ball into dangerous areas.
That continued into the second half as Burnley looked the more likely to find the net.
Kieran Trippier was a constant source of imagination down the right for the hosts and his cross on 61 minutes caused minor panic in the Tottenham box, with Vlad Chiriches inadvertently chesting the ball back to Vorm under pressure by Ashley Barnes.
Christian Eriksen blazed Tottenham’s best chance over the bar on 68 minutes, one of 10 attempts that the visitors hit off target during the course of the game.
Pochettino threw on Andros Townsend in the closing minutes but it was Burnley that carried the bigger threat.
With the clock ticking down, the hosts could have grabbed what would have been a huge goal in their battle to avoid relegation, but Boyd could only head wide from eight yards out.
Thierry Henry’s Super Sunday verdict:
“It’s a vital point for Burnley. If I had said to you at the beginning of the season, they would have a draw at Chelsea, drew just now against Tottenham, beat Man City and they have Arsenal coming soon, it’s a good result at the end of the day. I thought Danny Ings had a good game. He missed an opportunity at the beginning, but I thought he had a good game.”
Burnley: Heaton (7); Trippier (7), Duff (8), Shackell (7), Mee (7); Boyd (7), Jones (7), Arfield (6), Barnes (6); Ings (6), Vokes (6)
Subs used: Jutkiewicz (6)
Tottenham: Vorm (6), Walker (6), Chiriches (7), Dier (7), Rose (7); Bentaleb (6), Mason (5); Chadli (6), Eriksen (6), Paulinho (5); Kane (6)
Subs used: Davies (6), Lamela (6), Townsend (6)
Man of the Match: Michael Duff