Burnley have now dropped 10 points from winning positions this term as the Foxes came from behind twice to secure a point.
It was the fourth time that Sean Dyche's side had failed to hold on to a lead this season following results against Brighton, Leeds United and Everton.
The visitors had taken the lead twice in the first 45 minutes, firstly when Jamie Vardy put the ball into his own net from Ashley Westwood's corner.
The ex-England international made amends for that error in the 37th minute when firing across Nick Pope to equalise.
But Burnley were back in front just three minutes later when Maxwel Cornet met Matej Vydra's cross on the volley to beat Kasper Schmeichel at his near post.
The Ivorian's elation, however, soon turned to disappointment as he pulled up with a suspected hamstring injury and was replaced by Johann Berg Gudmundsson just before the break.
The Clarets defended for their lives after the interval and were on course to preserve their advantage, until Vardy intervened late on.
The striker raced on to a through ball, rounded Pope and slotted into an empty net to make it 2-2.
The away side thought they'd sealed their first win of the campaign in the fourth minute of stoppage time when Chris Wood headed in off the post.
Replays, though, showed that the New Zealander was a fraction off side and, following a brief VAR check, referee Chris Kavanagh ruled the goal out.