The attack occurred in Khuntapali, a village in Orissa state, during a strike called by the World Hindu Council to protest Saturday's killing of a Hindu religious leader and four others by suspected communist rebels in another district of the state, Ashok Biswal, superintendent of police, told The Associated Press.
Khuntapali is nearly 250 miles west of Bhubaneshwar, the capital of Orissa state.
In the past Hindu extremists in Orissa state have attacked Christian missionaries. In 1999 an Australian missionary, Graham Staines, and his two sons were killed by a Hindu mob that set their car on fire.
On Monday, a group of Hindu hard-liners converged on the orphanage and asked nearly 20 residents to leave the complex, Biswal said.
They then set the orphanage on fire with the nun and the priest locked inside, the police officer said.
The nun died and the priest was hospitalized with serious burns, Biswal said.
The region is marked by religious tensions between Christian missionaries who work with mostly poor tribes in the region and hard-line Hindu groups that claim the Christians are forcing or bribing people to convert.
Churches have denied that residents have been pressured or bribed to change their religious beliefs.
Indian law accepts missionaries but bars forced conversions. Nevertheless, any missionary activity generally provokes controversy.
Hindus account for 84 percent of India's more than 1.2 billion population and Christians about 2.4 percent.