The London club would have moved above Liverpool into fifth place with a win at Turf Moor on Sunday, but never really looked like securing all three points from a forgettable game that finished goalless.
Danny Ings wasted a great chance to put Burnley in front when he shot straight at Michel Vorm after only four minutes and that was as close as either side came.
A draw on the road ensured Spurs leapfrogged Southampton into sixth, but they look unlikely to secure a UEFA Champions League spot with seven games remaining.
Head coach Pochettino felt his side came up short in Lancashire, but urged them to put a lacklustre display behind them.
He said: "It was a very tough game. We fought a lot. The second half was good but not enough to win the game.
"I'm pleased with a clean sheet but we've dropped two points. We need to move forward and fight in the next games."
Harry Kane captained Tottenham for the first time in the absence of injured goalkeeper Hugo Lloris, but the striker was unable to add to his 29-goal tally for the season and gave credit to a well-drilled Burnley side.
"It's a huge honour and to lead the lads out was a very proud day for me just a shame we couldn't get the win," he commented.
"They made it tough for us. It's another point gained but we were looking for all three today.
"A clean sheet away from home is always pleasing so credit to the defence."
