Something that "burns a chill" would be a scenario or event that would also kill a buzz or harsh a mellow. Such things include: loud noises, over-excited or un-chill people, running out of the means to be chill, eating the last cookie, etc. "burning a chill" is just another saying in accordance to harshing a mellow or killing a buzz.
Quite simply, a chill burn is a buzz kill.
"Hey man why don't we get off our asses and go out and do something fun?!?!"
"Dude, that would totally burn my chill"
"Spence, it's time to go shopping with me; you promised!"
"Relax, brosephine, you're such a chill burn."