Both sides created chances to score in Tuesday night's Championship clash at the King Power Stadium, but Charlie Austin came closest to breaking the deadlock just before the hour mark when he saw a header strike both posts.
A magnificent tackle from Leicester defender Sol Bamba then prevented the Burnley striker from tapping the ball into an empty net from two yards out.
The Foxes' best chance was shortly before that moment of drama as Kieran Trippier cleared a header from Lee Peltier off the line.
Burnley's play-off hopes were all but over before kick-off and more dropped points virtually ends Leicester's top-six challenge as well.
Both sides made two changes from the weekend.
Danny Drinkwater, Leicester's goalscorer in the defeat at Millwall, started while Lloyd Dyer was also recalled. Andy King and Paul Gallagher dropped down to the bench.
Danny Lafferty and Marvin Bartley came into the Burnley team as Josh McQuoid and Danny Ings began as substitutes.
Dyer almost made an immediate impact.
A clever drop of the shoulder and turn of pace took him between two defenders and into the penalty area but goalkeeper Lee Grant blocked his low shot with barely a minute on the clock.
At the other end, Kasper Schmeichel had to beat away a swerving first-time shot from Austin.
An under-hit backpass from Bamba allowed Austin a clear run at goal in the 15th minute.
Leicester looked in trouble but the Burnley striker then scuffed his cross and Paul Konchesky nipped in front of intended target Chris McCann to clear the danger.
Austin miscued again soon afterwards when he should really have worked Schmeichel after getting across his marker and on the end of Lafferty's near-post cross.
Grant got down to smother Drinkwater's 25-yarder midway through the first half before Konchesky whipped a dangerous, low free-kick into a crowded penalty area just after the half-hour mark, but once again the former England Under-21 keeper averted the threat.
Grant was a mere bystander though when Drinkwater's flicked header from Ben Marshall's cross looped beyond him and dropped wide of the far post just before half-time.
Leicester continued to probe at the start of the second half and came close to scoring when Peltier met Marshall's corner, but Trippier cleared off the line.
Grant was worked again shortly afterwards, blocking with his feet after Jermaine Beckford's flick inside the area released David Nugent.
Austin came closer than anyone to breaking the deadlock just before the hour mark.
The former bricklayer rose above Bamba from Lafferty's cross.
His header beat Schmeichel but not the far post. The ball then bounced along the goal-line and hit the other post.
Austin looked certain to tap into an empty net from two yards out but Bamba somehow got back and slid in to block his follow up, happy to concede a corner.
That was the end of the chances until Drinkwater unleashed another long-ranger in the 83rd minute - this one whistled narrow past the top corner.