Kumar was 98 years old and had been suffering for some time, one of the doctors treating him told reporters.
Prime Minister Narendra Modi and his Pakistani counterpart Imran Khan expressed their condolences, Modi saying Kumar was “blessed with unprecedented brilliance”.
“For my generation, Dilip Kumar was the tallest and most versatile actor,” Khan said on Twitter.
At his residence in Mumbai, Bollywood actors and filmmakers including Shah Rukh Khan, Vidya Balan and Karan Johar flocked to pay tribute to him.
The funeral was held with state honors at a cemetery in Mumbai.
Authorities in Peshawar, Khyber Pakhtunkhwa province, said they plan to restore the actor’s ancestral home in a narrow alley in town.
“We are now working on its conservation,” said Abdus Samad, director general of the archeology department.
“An institution has disappeared … every time the history of Indian cinema is written, it will always be before Dilip Kumar and after Dilip Kumar,” actor Amitabh Bachchan said on Twitter.