Forward Alexandre Lacazette went closest for the Gunners, poking an Emile Smith Rowe cutback narrowly wide midway through the second half.
Smith Rowe also forced a fine diving stop from Burnley keeper Nick Pope in the second period while Martin Odegaard was denied by the stopper’s feet with a powerful effort in the first half.
A draw for Burnley ended a run of three straight defeats with the Clarets now four points off safety having played four games fewer than 17th-placed Norwich City.
Meanwhile Arsenal, now winless in their past five matches in all competitions, go sixth, two points off Manchester United in fourth.
Arsenal booed off after poor start to 2022
Arsenal will be pleased to see the back of January, so far in 2022 the Gunners have been dumped out of the EFL and FA Cups and have lost three of their past four matches.
Arsenal dominated possession but were frequently frustrated in the first half by their opponent’s low-block.
After the break, the hosts upped the tempo, but Bukayo Saka, Smith Rowe, Odegaard and Lacazette all failed to hit the net despite finding themselves in promising positions.
This season appears to be Champions League qualification or nothing for Mikel Arteta’s team, making their dropped points against their injury-hit opponents all the more infuriating for supporters.
A small section of home fans booed their team off at full-time, the five-game winning streak at the end of 2021 an increasingly distant memory.
Burnley battle to much-needed point
Burnley’s game plan was obvious from the first whistle, they defended as a unit, sitting behind the ball and stifling their opponents’ attacks.
This was the Clarets’ first match since losing to Huddersfield Town in the FA Cup on 8 January with Sean Dyche’s side seeing league matches against Leicester City and Watford postponed in recent weeks.
Since they last played, forward Chris Wood has moved to relegation rivals Newcastle.
Jay Rodriguez and Matej Vydra started up front together in the target-man’s absence, but the pair rarely received the ball in attacking positions.
With Maxwel Cornet representing Ivory Coast at the Africa Cup of Nations, defender Ben Mee was Burnley’s leading Premier League goalscorer among the starting XI with two goals scored this season.
And it was Mee who had the visitors’ best opportunity, volleying a half-chance off-target from a corner in the first half.
Dyche will have been delighted with the defensive performance of his side – their next fixture, at home to fellow relegation-candidates Watford, will be one in which a more attacking mindset will surely be employed.