"Since being spread by Hagoromo, better known as the Sage of Six Paths, chakra has become a form of life energy that all individuals produce to some degree;
those who run out of chakra will die. Produced within the "chakra coils" that mainly surround and connect to
each chakra-producing organ, the energy circulates throughout the body in a network called the "Chakra Pathway System", which is similar to the cardiovascular system."
From the wikia with highlighted parts. There's 8 organs that produce chakra. See: The 8 Gates.
However I do believe the heart produces at least 90% of all chakra. Emphasis has been shown in the show on how important the heart is. 8th gate is there, Sasori still needed his heart, Kakuzu needed one, and so on. Seems very important to have.
I'm in the section where, without a heart, immortal or not, you'll eventually run out of chakra. Dying isn't part of that as Hidan is truly immortal. Unless you want to cut him up and wait 100 years for his body to decay from lack of nutrition.
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Good luck getting a consensus from this.