Maurizio Sarri's men were hoping to bounce back from a 2-0 defeat to Liverpoolin their previous top-flight outing but they dropped points once again and could be facing key games against Manchester United and Eintracht Frankfurt without Kante.
Burnley – still yet to confirm their survival – went ahead through Jeff Hendrick early on, and although Kante and Gonzalo Higuain saw Chelsea turn things around in the space of two minutes, Ashley Barnes pegged them back in an enthralling first half.
Chelsea did much of the attacking after the interval but Sarri – who was removed from the sidelines in stoppage time – saw his side fail to make the most of top-four rivals Tottenham, Arsenal and United losing at the weekend.
Burnley edged ahead just eight minutes into a thrilling first half when Hendrick sent a fine volley into the bottom-right corner, but the Blues swiftly equalised thanks to Kante's left-footed effort into the roof of the net after fine work from Eden Hazard.
A brilliant move in the 14th minute saw Higuain score for the first time since March 3, rifling in off the crossbar from a tight angle following Cesar Azpilicueta's clever flick.
However, Burnley restored parity 10 minutes later when Barnes latched on to Chris Wood's glancing header.
Having lost Hudson-Odoi late in the first half to an apparent shin problem, Kante was unable to come out for the second period due to a reported back injury.
Chelsea still looked cohesive going forward and nearly retook the lead in the 51st minute, but Tom Heaton thwarted Higuain – who reacted furiously when he was replaced by Olivier Giroud in the 77th minute.
The hosts piled the pressure on towards the end but failed to craft any further opportunities and Sarri's late ejection for dissent and a fracas after the final whistle – which was met with boos – summed up Chelsea's day.
Top-four race tighter than ever
A victory would have put Chelsea on 69 points, two clear of Tottenham, three ahead of Arsenal and five away from United.
With their rivals all in action on Tuesday or Wednesday, Chelsea could instead find themselves in an even tighter tussle if results do not go their way.
