Burns made the announcement one day after the defending Stanley Cup champions were eliminated from the playoffs by Philadelphia Flyers in Game Five of their Eastern Conference quarterfinal series.
"For those of you who know me well, I've never backed down from any fight and I'm not going to back down from this one," Burns said.
Burns read from a prepared statement and did not answer any questions from the media.
No announcement was made on the future status of the Devils' coaching situation.
The 52-year-old Burns missed a morning skate and an off-day team meeting this week for "undisclosed personal reasons."
A three-time winner of the Jack Adams Trophy as NHL Coach of the Year, Burns won his first Stanley Cup in his first year as Devils coach in 2002-03.