Saber, 52, died in a Tehran hospital after being transferred from Evin prison, his sister Firouzeh told Radio Farda, the Persian-language service of Radio Free Europe/Radio Liberty.
The family is charging that authorities waited several hours after Saber complained of chest pains to hospitalize him.
He began his fast June 2 to protest the death of Haleh Sahabi, who was killed June 1 at the funeral of her father, another dissident.
Sunday is the second anniversary of the disputed presidential election that re-elected Mahmoud Ahmadinejad, and police attacked hundreds of protesters, Israel's Haaretz newspaper reported, citing Iranian opposition Web sites.
Saber's wife, Farideh Jamshidi, told Britain's The Guardian newspaper: "My husband died two days ago, but we were unaware of his death until today, when someone in the hospital informed one of our friends.
"Doctors told us later that they could have saved his life by taking him to the hospital earlier. We were supposed to visit him in the prison tomorrow, and now we have to visit his dead body in the cemetery."
The semiofficial Fars news agency denied the allegations.
Notable deaths of 2014 [PHOTOS]