The Foxes were one of the early-season favourites for promotion due to their activities in the transfer market, but haven't performed as many had anticipated.
Despite this they were still in with an outside shot of breaking into the top six prior to kickoff, but a stalemate against the Clarets ends their ambitions of Premier League football next season.
The match was an even contest, with the visitors having the better of the first 45 minutes. Kieran Tripper shot just wide from the edge of the area and Ross Wallace came even closer to opening the scoring, but his free kick was saved by Kasper Schmeichel.
Nigel Pearson's hosts grabbed control of the game in the second half and created several chances of their own, but couldn't find a break in the deadlock.
The result leaves Leicester in 10th place with 63 points, while Burnley are one place and two points behind them in the Championship table.