The runaway Premier League leaders made a great start at Turf Moor and were in front in just the seventh minute when Pedro completed a fine counter-attack.
That proved to be Chelsea's last shot on target, though, and the Blues were, fittingly, caught cold on a dour day in Lancashire as Robbie Brady marked his first Burnley start with a stunning free-kick to equalise.
Chelsea huffed and puffed, but were met by a resilient Burnley defence and the Clarets might have won the game had Matt Lowton or Andre Gray shown more composure.
Speaking to Sky Sports, Chelsea boss Conte said: "We must be disappointed for this point because I think they stopped our game. We could have won, it's a pity because we started really well and created chances to score goals and then Burnley found the draw with the free-kick.
"But now it's important to re-start. It's a draw, we take a point but we must be a bit disappointed.
"It's a pity because we started very well, we kept the situation under control and also created many chances to score the goal in this time, maybe we can kill the game but it didn't happen.
"Then [they scored] a free-kick, then second half it became a physical game with the long ball.
"We don't know this type of game, the long ball is not easy to play here with Burnley. They took 29 points here now after this game, from 30 points in the table.
"At home for sure they're a strong team, but I think we're disappointed not to win."
With a draw, Chelsea move 10 points clear of nearest rivals Arsenal and Tottenham at the top of the table, but Conte says nothing has changed in terms of what is required to bring the title back to Stamford Bridge.
"For sure, we already know this [that Chelsea have to work hard]," he added.
"Now there are 13 games. We know the way is a long way to go, but it is important to continue to work in this way, to play this type of game, to fight."