Clarets manager Eddie Howe missed the game due to the death of his mother so assistant Jason Tindall took charge of the side and the Clarets were in front after two minutes through Jay Rodriguez's penalty after Jonathan Parr had fouled Martin Paterson in the box.
The home side had goalkeeper Lee Grant to thank for keeping them in front as he made good saves to deny Glenn Murray and Paddy McCarthy before the break, but Palace levelled in the 64th minute.
Wilfried Zaha was brought in the box by Kieran Trippier after a clever run and Darren Ambrose stepped up to level from the spot.
Zaha saw red with 18 minutes remaining for an off-the-ball clash with David Edgar but Burnley were unable to find a winner and remain eight points adrift of the play-off positions.
The game was barely a minute gone when Parr and McCarthy got in a mix-up and Paterson was brought down in the box by Parr. Referee Dean Whitestone had no hesitation in pointing to the spot and Rodriguez hammered in his 21st goal of the season.
Burnley were dominating and Danny Ings, who was making his full debut for the Clarets, danced his way down the left and into the penalty area but Palace goalkeeper Julian Speroni was quick off his line to make a good save.
Palace came close to an equaliser nine minutes before the break when Ambrose crossed to the back post and Murray headed goalward but Grant made a superb flying save.
With half-time approaching Grant tipped a shot from Chris Martin round the post and the resulting corner from Ambrose found the head of McCarthy, but Grant was on hand to make another good save.
Palace continued to push forward after the break and Murray had another header saved by Grant before Zaha was cynically brought down by Michael Duff and Ambrose fired a free-kick straight at the Burnley stopper.
The Burnley defence were struggling to cope with Zaha and just after the hour mark the England Under-21 international was fouled in the box by Trippier and Ambrose scored the penalty.
Trippier came close to making amends soon after but his free-kick was tipped over the bar by Speroni.
Zaha had enjoyed another productive afternoon but it ended in misery when he was shown a straight red card for lashing out at Edgar.
Burnley pressed for a winner but Ings headed over and they were forced to settle for a point as their winless run extended to four matches.