Victory would have seen the Foxes edge into the top six ahead of Saturday's fixtures, and it looked like Nigel Pearson's side were going to end their eight-game winless run after Jeff Schlupp had headed them back in front. But substitute Burke, back after a three-game absence with an ankle injury, kept his cool from the penalty spot to snatch a point for the visitors.
Curtis Davies' own goal saw Leicester take an early lead but Nikola Zigic equalised just after the hour mark. The result leaves the Foxes in seventh place with only one win from their last 13 league matches.
The opening goal came following a swift counter attack. Wade Elliott's corner was cleared by Danny Drinkwater and controlled brilliantly by Chris Wood inside the centre circle while under pressure from a couple of defenders.
Wood slipped a pass into the path of David Nugent whose 25-yard effort had to be turned behind by Jack Butland. From the resulting corner, Matty James' cross was headed in by the stooping Davies as he attempted to prevent Wes Morgan from getting to the ball.
Leicester were on top and another James corner was met by Michael Keane on the run but the on-loan Manchester United centre-half powered his header narrowly over the crossbar from eight yards out. Andy King then created a chance for himself after chasing down his own header behind the Birmingham defence but Butland smothered his shot.
Elliott then curled a left-foot effort narrowly wide of the far post from the right hand-side as Birmingham continued to have their moments. And they had a moment that really mattered after 61 minutes, scoring a well-crafted equaliser. Nathan Redmond and Paul Caddis carved Leicester open down the right, with the latter providing a low cross which Zigic tapped in from six yards.
Schlupp was the width of a post away from restoring the host's lead shortly afterwards when his fierce left-foot shot struck the upright. The Ghanaian did put Leicester back in front after 75 minutes when he rose unmarked to head home James' corner.
It looked to be enough for Pearson's side until the ball bobbled back into the area off Nugent, who then tripped substitute Wes Thomas.
The referee pointed to the spot and Burke stepped up to level it up once more.