The 16th Century mosque in Ayodhya town was razed in 1992 by a Hindu mob who want to build a temple to Lord Ram.
The threat was made by T Raja Singh, a senior leader of India's governing BJP party in Telangana state.
A BJP spokeswoman criticised the remarks Mr Singh made on 5 April.
Late on Sunday, police in Hyderabad city registered a case against him for "deliberate and malicious acts intended to outrage religious feelings of any class by insulting its religion or religious beliefs", after a complaint by a Muslim political organisation.
Mr Singh has defended his statement on the flashpoint issue, saying it was aimed at those who were opposed to the Ram temple in Ayodhya.
"For the construction of the temple, we can sacrifice our lives and even kill others, this is my warning to those who want to create hindrance in the construction of the temple," Hindustan Times quoted him as saying.
In the past too, Mr Singh has courted controversy when in December 2015 he supported the lynching of a Muslim man by a Hindu mob on suspicion that he had stored beef in his home in the northern state of Uttar Pradesh.
Last year, he justified the beating of some Dalit (formerly known as untouchables, who form the lowest rung of India's caste hierarchy) men by cow vigilante groups in Gujarat.
Police in southern India have filed a case against a legislator who threatened to behead those opposed to the building of a Hindu temple on the site of a destroyed mosque.