Kishore was a senior leader of the Vishwa Hindu Parishad (VHP), also called the World Hindu Council, and died at the party's headquarters.
He began working for VHP in 1983 and served as its vice president.
Kishore was credited with pursuing the establishment of a temple to honor the Hindu god Ram in the northern town of Ayodhya and, in the same town, he was also accused of inciting a Hindu mob to destroy a 16th century Babri mosque.
Bharatiya Janata Party chief Amit Shah lauded Kishore for his commitment to the Hindu cause.
"Acharya Giriraj Kishore served the society and the country through RSS, Jan Sangh and VHP till his last breath. He left his job as a teacher to dedicate his life for the cause of Lord Ram. He fought against Emergency and went to jail...
A symbol of sacrifice and fearlessness, he donated his body for the social cause. He will remain a source of inspiration for people."
Kishore had been suffering from an extended illness prior to his death, the Times of India noted.