Burdonsky was born Alexander Stalin but took his mother's maiden name to escape association with the Soviet dictator.
For several years he was head of the Central Russian Army Theatre in Moscow.
Last year another grandson of Stalin, Yevgeny Dzhugashvili, died aged 80. However, he was a staunch defender of his grandfather's legacy.
Alexander Burdonsky's father, Vasily Stalin, had a difficult relationship with the Soviet leader and died an alcoholic aged 40.
Born in 1941, Burdonsky told Russian TV that he had little to do with the Soviet dictator before his death in 1953: "I saw him during the May and November parades from the stands. I only saw him close-up... lying in state."
Two years ago the theatre director said that nobody would want the childhood he had had, and that he had changed his name to pursue a life in the arts.
He once told an interviewer that he saw his grandfather as a Shakespearean character, a mixture of genius and madness. While he had been bitterly opposed to him as a younger man, he said he had come to see Stalin as a very powerful 20th-Century figure, despite his tyranny and cruelty.
Yevgeny Dzhugashvili, who died in December 2016, saw Stalin rather differently. When an independent Russian newspaper described his grandfather as a "bloodthirsty cannibal", he took the paper to the European Court of Human Rights and lost.
Alexander Burdonsky, a talented Russian theatre director whose grandfather was Soviet leader Josef Stalin, has died of cancer aged 75.