Claudio Ranieri's side ended 2015 level on points with leaders Arsenal, despite finishing the year with a defeat at Liverpool and another home stalemate against Manchester City.
They dropped more points on Saturday despite playing for more than half an hour against 10 men following Simon Francis' red card
That incident led to Leicester winning a penalty but Mahrez saw his spot-kick saved by Artur Boruc.
Ranieri said this week he felt the Bournemouth game would be tougher than City's visit, and he was not far wrong
The Foxes huffed and puffed and Jamie Vardy failed to find the back of the net for a fourth successive match as Leicester fell two points adrift of Arsenal at the Premier League summit
Mahrez had much of the King Power Stadium on its feet in the 10th minute when his spectacular overhead volley flew just over the crossbar
Shortly afterwards Boruc was quickly out of his goal to gather at Leonardo Ulloa's feet and deny the Leicester striker
Bournemouth, now beaten just once in their last seven matches, saw a big chance go begging after quarter of an hour when Josh King glanced his header wide after Junior Stanislas had picked him out inside the six-yard box.
It was Leicester's turn to miss a sitter as the half-hour mark approached
Leading scorer Vardy controlled a mis-hit shot from Ulloa around the penalty spot but with only Boruc to beat he poked his shot against the inside of a post and the ball rebounded away.
Stanislas and Dan Gosling were then guilty of scooping shots over the crossbar from good positions inside the penalty area.
Back at the other end, Mahrez put in Vardy down the left-hand side of the box but the 15-goal striker missed the far post with his low effort from a tight angle.
Further scoring opportunities came and went for Leicester at the beginning of the second half as a fingertip save from Boruc denied Vardy before captain Wes Morgan volleyed the ball over the crossbar from inside the six-yard box following a corner.
For the first time in the match the momentum was firmly with one team and the pendulum swung further in Leicester's favour in the 57th minute with the dismissal of Francis.
The Bournemouth captain and Vardy chased after Danny Drinkwater's through ball, and with Vardy winning the race, Francis made a sliding challenge, bringing down the Foxes man
Television replays suggested Francis got a touch on the ball, and he protested as much, but referee Andre Marriner brought out his red card and awarded the penalty.
Bournemouth will feel some justice was served as Boruc, diving to his left, got two strong hands to Mahrez's spot-kick.
Eddie Howe's team responded well to their numerical disadvantage
The introduction of experienced defender Sylvain Distin, for midfielder Harry Artur, helped as Bournemouth by and large kept Leicester at arm's length for the remainder of the game.
Leicester did have some chances, though
Substitute Shinji Okazaki sent a volley narrowly wide before Vardy glanced a header past a post in between two efforts by N'Golo Kante from outside the box.
Ranieri's team were to be frustrated though and are now without a win in three games.
TWEET OF THE MATCH
@GaryLineker: "Not quite sure how Leicester didn't win, but credit to Bournemouth for a mightily spirited performance."
PLAYER RATINGS
Leicester:
Kasper Schmeichel: 6
Danny Simpson: 6
Wes Morgan: 7
Robert Huth: 7
Christian Fuchs: 6
Riyad Mahrez: 5
N'Golo Kante: 7
Danny Drinkwater: 7
Marc Albrighton: 6
Leonardo Ulloa: 5
Jamie Vardy: 6
Substitutes:
Nathan Dyer: 6
Shinji Okazaki: 6
Ritchie De Laet: 6
Bournemouth:
Artur Boruc: 8
Simon Francis: 6
Steve Cook: 7
Dan Gosling: 7
Andrew Surman: 7
Harry Arter: 6
Charlie Daniels: 7
Adam Smith: 6
Matt Ritchie: 6
Josh King: 7
Junior Stanislas: 7
Substitutes:
Glenn Murray: 7
Sylvain Distin: 7
STAR MAN
Artur Boruc - Made a good save to deny Vardy before keeping out the penalty
Organised his defence well and was always talking, especially when his team went down to 10 men
Earned his clean sheet.
MOMENT OF THE MATCH
Francis' red card
It was a big call from the referee and it had a profound affect on the remainder of the game, although without it we would not have had the penalty drama involving Mahrez and Boruc.
VIEW FROM THE BENCH
Ranieri will be disappointed his team could not break down 10 men and keep the pressure on Arsenal
Of more concern to the Italian though will be that the goals have dried up for his free-scoring team
Leicester have drawn a blank in their last three matches.
Eddie Howe will be delighted given the circumstances
Bringing on Distin proved a masterstroke after the red card and his experience and organisational skills were evident
Another good point for the Cherries.
MOAN OF THE MATCH
The penalty call
Such a big decision and replays suggested Andre Marriner got it wrong
But without the help of technology for such decisions, mistakes will continue to be made.
WHO'S UP NEXT?
Tottenham v Leicester (FA Cup, Sunday January 10)
Birmingham v Bournemouth (FA Cup, Saturday January 9)
Source : PA
Source: PA