Jeff Hendrick put the Clarets into an early lead, before N'Golo Kante and Gonzalo Higuain struck back for the hosts. Burnley answered back again when Ashley Barnes headed in after 24 minutes.
The stalemate leaves the Blues trailing third-place Tottenham Hotspur on goal difference. Chelsea lead Arsenal by a point and Manchester United by three but have played a game more than their nearest rivals in the race to qualify for next season's UEFA Champions League.
No One Wants Top 4
Fourth place in the Premier League isn't a hot potato, it's a ticket to the lucrative Champions League. Still, you could be forgiven for thinking the opposite based on how Chelsea, Arsenal and Manchester United approached must-win games in the top-four race this weekend.
Chelsea turned up to host a goal-shy Burnley outfit and promptly conceded twice while barely twitching a hair. Jeff Hendrick's eighth-minute strike was a peach of a hit, but he had the freedom of west London to make it.
Equally sweet strikes from Kante and Higuain should have been the nerve-settlers Chelsea needed. Instead, the Blues remained generous at the back, allowing Ashley Barnes to head in under no pressure.
It was hardly surprising manager Maurizio Sarri sought an inquest into what was going wrong defensively. Wisely, Sarri chose David Luiz, no stranger to an error, as the man to provide him with answers.
The discussion hardly bore fruit since Burnley started the second half on the front foot, creating chances with ease. Those chances didn't yield goals only because of the Clarets' own profligacy, rather than any Chelsea improvement.
What Chelsea’s shaky displayed proved beyond doubt is the winner of the race for fourth will likely be the least incompetent group standing.
If Chelsea fans seek a consolation, it comes from the hapless performances their nearest rivals had already produced.
A trip to Everton was always going to represent a tricky fixture for United. Yet the manner of the surrender during Sunday’s 4-0 defeat has to have alarm bells ringing around Old Trafford ahead of the visit of derby rivals Manchester City on Wednesday.
United hardly look Champions League-worthy thanks to a defence more vulnerable than Chelsea's suspect unit.
Then there's Arsenal, a club where defensive lapses are so common it's as if they are contractual obligations for every player. Step forward Shkodran Mustafi, who was chiefly responsible for all three of the goals Crystal Palace scored at the Emirates Stadium on Sunday.
The Eagles have been strong on their travels, but the Gunners had turned their home into a fortress of sorts. Quite how Arsenal failed to translate the confidence accrued from beating Napoli in the UEFA Europa quarter-final into a home banker against a team entering Sunday still not free from the threat of relegation, is beyond baffling.
Maybe Mustafi and the Gunners were just victims to the malaise afflicting the three teams scrapping for fourth. No one member of the trio looks good enough to deserve it.
Chelsea face United at Old Trafford on Sunday after Burnley host title-chasing Manchester City at Turf Moor.