The bishop is under investigation for alleged extortion of priests in his diocese, and the covering-up of sexual abuse cases, according to Brazilian media reports.
Dias de Oliveira allegedly protected a priest accused of clerical abuse.
The bishop confessed to police in April that he had swiped $4,000 (3,580 euros) from parish funds for personal use, telling them he did so because he was having financial difficulties, the reports said.
The Vatican always announces when Francis has accepted the resignation of a bishop, but never explains why.
"Dear brothers and sisters, in recent months we have had to bear all sorts of crosses through attacks against our Limeira church, against me and other presbyters," Dias de Oliveira said in a statement.
"I acknowledge my limitations, but I also carry in my heart all the love I have received from the good people of God.
"I ask to resign for the love of Christ and for the good of the diocese," he said in the letter published by the diocese of Limeira, a city in the Sao Paulo state in southern Brazil.
The Catholic Church has been rocked by a global clerical paedophilia scandal, with victims coming forward in countries ranging from Australia to Chile, Germany and the United States.
Pope Francis passed a landmark new measure this month to oblige those who know about sex abuse in the Church to report it to their superiors, in a move which could bring countless new cases to light.
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