Hoeness, one of the most prominent figures in German football, was sentenced to three years in prison after being convicted of evading millions of euros in tax through an undeclared bank account.
"After discussions with my family, I have decided to accept the decision of the Munich state prosecutors regarding my tax issues," Hoeness wrote in a statement on Bayern's website.
"I have asked my solicitors not to appeal the decision. That reflects my views on decency, conduct and personal responsibility.
"Tax evasion was the biggest mistake of my life and I will face up to the consequences of this mistake.
"I will also be resigning from my position of president of Bayern Munich. By doing this, I would like to prevent any damage to my club.
"Bayern Munich are my lifetime's work and they will remain such. I will remain attached to this amazing club and its people in a different way for as long as I live.
"I would like to thank all my personal friends and the supporters of Bayern Munich from the bottom of my heart for their support."
The former West Germany international handed himself in to authorities earlier in March after confessing to evading 18.5m euros in taxes by hiding his income in various Swiss bank accounts.
Prosecutors had initially been targeting a five-year sentence, half of the maximum punishment for major tax fraud under German law.
Bayern moved quickly to fill the vacancy left by Hoeness' departure with adidas chairman Herbert Hainer elected as his replacement.
Hainer, 59, will assume Hoeness' duties after the club accepted his earlier resignation.
Hainer received a majority vote in an emergency board meeting on Friday and assumes his new position with immediate effect.
"Uli Hoeness has always served Bayern Munich's best interests with his leadership qualities, his high personal involvement and his exceptional lifetime achievements," Hainer said in a statement.
"He has made a considerable contribution to Bayern Munich becoming one of the most successful and attractive clubs in the world, financially and sportingly, and we will always be extremely grateful to him for this."