Adnan Hassanpur and Hiwa Butimar were reportedly sentenced in the northwestern Kurdistan Province on July 16.
Hassanpur's sister told Radio Farda that the men launched their hunger strike to protest their sentences.
"They have entered their [39th day] of hunger strike," she said. "Their lawyers, who were able to meet them for the first time [recently], have said that their psychological and physical conditions are a cause of concern."
Reports have suggested the two men had contacts with Iranian armed opposition groups.
Judiciary officials have said the two were involved in an "armed struggle" against Iran's Islamic establishment.
"We are seriously concerned and we are calling on all international humanitarian organizations not to remain silent and send a team of doctors to save the lives of [Hassanpur and Butimar]," Hasanpur's sister told Radio Farda.
Amnesty International has called on Iranian authorities to commute Hassanpur and Butimar's death sentences.
Media rights groups said Hassanpur had writen articles on local politics, while Butimar was a member of an environmental group.