In calls made to journalists in northwest Pakistan, Tehrik-e-Taliban Pakistan spokesman Ihsanullah Ihsan said his group would target Indians and Indian interests "anywhere" to avenge the hanging of Kasab.
Ihsan further said Indian authorities should hand over Kasab's body to his family or to the Taliban.
He warned the Taliban would "capture Indians" and "not return their bodies" if Kasab's body was not handed over.
25-year-old Kasab, who belonged to Pakistan's most populous province of Punjab, was executed in a Pune jail early yesterday.
His body was buried within the prison.
The Pakistan government reacted cautiously to the hanging, with the Foreign Office spokesman saying Islamabad was against all forms of terrorism and would cooperate with countries in the region to tackle the menace.
Kasab was the only surviving member of a 10-member team of Lashkar-e-Taiba (LeT) terrorists who killed 166 people during a three-day rampage in Mumbai nearly four years ago.
Diplomatic sources said security measures at the Indian High Commission, which are already very extensive, had been further tightened after the Taliban threat.
Indian diplomats in Pakistan have faced threats in the past too.
In 2010, two deserters from a paramilitary force arrested for terror links had revealed that militants aligned with the Pakistani Taliban had plans to abduct Indian diplomats and target foreign missions.
The men had revealed that militant groups had plans to kidnap Indian diplomats, including the High Commissioner, to exchange them for militant commanders being held by Pakistani authorities.
At that time too, security had been tightened for Indian diplomats and their families.
Yesterday, an unnamed LeT commander was quoted by media as saying that the group regarded Kasab as a hero who would inspire more terror attacks.
Ihsan was quoted by the Pakistani media yesterday as saying that the Taliban were shocked by Kasab's hanging.