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Aston Villa also missed the chance to move to within two points of the top four as they suffered a shock 1-0 defeat by bottom-of-the-table Sheffield United, who held on for the final half an hour with 10 men.
Leicester's challenge for a return to the Champions League next season has been badly hit by a series of injuries to key players and the presence of James Maddison, Harvey Barnes, Jonny Evans and James Justin was badly missed.
A point edges Brendan Rodgers's men level on points with second-placed Manchester United, who face Crystal Palace later on Wednesday.
But Chelsea, Liverpool and Everton all have the chance to close in on Leicester -- who have not won in their last three games -- when they are in action on Thursday.
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A newly configured defence and midfield for the visitors showed as Hamza Choudhury's slack pass to Wilfred Ndidi allowed Matej Vydra to smash home his first league goal in nearly 12 months after just four minutes.
Choudhury missed a great chance to quickly make amends as his looping effort was tipped over by Nick Pope.
However, Kelechi Iheanacho's splendid finish drew Leicester level on 34 minutes as the Nigerian volleyed home from Ndidi's floated ball over the top.
Schmeichel then produced two stunning saves to prevent Chris Wood restoring Burnley's lead early in the second half.
The Dane leapt to his left to parry the New Zealander's powerful header before getting down low to his right to turn another deflected Wood shot wide.
Ashley Westwood then rattled the post for Burnley, but Leicester finished the stronger and could have snatched a victory that would have taken up to second in the table.
Marc Albrighton stung the palms of Pope before Youri Tielemans's shot came back off the post.
A point takes Burnley six points clear of the bottom three in 15th.
Sheffield United secured just their fourth win of the season as David McGoldrick scored the only goal against a Villa side desperately missing the inspiration of injured captain Jack Grealish.
McGoldrick swept home George Baldock's cross off the underside of the bar.
Bertrand Traore fired wide from the edge of the area and Ollie Watkins hit the post for Villa before Phil Jagielka was sent off for denying a clear goalscoring opportunity when he chopped down Anwar El Ghazi.
Despite dominating possession, Villa failed to make the man advantage count to remain down in ninth.
Blades boss Chris Wilder has already conceded his side are headed back to the Championship, but they close the gap to survival to 12 points.
kca/jc