The Briton, 28, has admitted taking cocaine to deal with depression and could also lose his boxing licence on Thursday.
He has not fought since beating Wladimir Klitschko in November 2015 and has twice withdrawn from rematches.
"I now enter another big challenge in my life which I know, like against Klitschko, I will conquer," Fury said.
He said it was "for the good boxing" and "only fair and right" to give up his belts.
"I won the titles in the ring and I believe that they should be lost in the ring, but I'm unable to defend at this time and I have taken the hard and emotional decision to now officially vacate my treasured world titles," he said.
In a statement, Fury's promoter Hennessy Sports said the decision would "allow him the time and space to fully recover from his present condition without any undue pressure and with the expert medical attention he requires".