The Clarets let slip a two-goal lead but still extended their unbeaten league run to nine games to stay in touch in the play-off race.
Rhys Fenlon put Burnley ahead in the opening five minutes from a well-worked free-kick before Chris Conn-Clarke doubled the lead with just over half an hour gone as the Clarets dominated before the break at Palace’s Academy training ground.
But the Eagles hit back in the second half to snatch a share of the points thanks to goals from a trialist and Branon Aveiro to take Burnley’s destiny out of their hands.
Tony Philliskirk’s side remains third in the Professional Development League’s northern division with four games to go.
But leaders Leeds United and second-placed Sheffield Wednesday both won, meaning they must drop points in the run-in to keep the door open for the Clarets to claim a top-two finish and a play-off berth.
Burnley: Allen, Connelly, Rain, Woode, Pruit, Conley, Fenlon, Conn-Clarke (Chima 90), McGlynn (Major 70), George, Steels (Tucker 72). Substitute: Waller.