The Wire promulgated the phone meaning of burner AFAIK. It may have been misunderstood by the police (Ed Burns was a police officer) or it may have been by analogy.
(I learned this from articles over a decade ago, very hard to pin down now that the phone meaning is widespread.)
"The writers of The Wire did not invent this term. Evidence of its use can be found at least as far back as 1996, when the rapper Kingpin Skinny Pimp used it on the song One Life 2 Live: “Talkin’ on the burner phone, bumpin’ hutch.”"