Tanasijevic, 75, came under strong pressure from fans to step down after the club revealed that it owed around 23 million euros ($30 million) to its creditors.
"I did not quit because of the fans," Tanasijevic was quoted as saying by Belgrade's B92 broadcaster. "They are not just Red Star's problem, they are the government's problem too."
Tanasijevic's resignation came after the die-hard section of Red Star fans said they would take "radical measures" to force the board out following the club's 1-0 league defeat to Javor Ivanjica on Wednesday.
The result left Red Star 20 points behind city foes Partizan and ended their title challenge with six games remaining after a 17th successive exit from Europe before the winter break.
Last month, the club had its water supply and telephone lines cut off for days over unpaid bills while players complained they had not been paid for three months.
Some of them said they had no money for food or rent and the gravity of the situation became all too apparent on Wednesday, when the team were forced to travel to Ivanjica on match day because they had no money for a night in a hotel.
"We've had an outstanding relationship with Tanasijevic, who worked hard to get the club out of the doldrums," said managing director Ilija Ivic.
"He came close to producing good solutions for the club's stabilisation, but this has proved to be an uphill battle."
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