Devendra Prasad Sinha was kicked and beaten with sticks by a group of men in Nalanda district on Sunday.
The incident happened after the bodies of two students who attended Mr Sinha's private school were found in a canal.
Villagers accused teachers from the private school of killing the students. No arrests have been made.
The mob attacked Mr Sinha after the bodies of Ravi Kumar and Sagar Kumar were recovered from a canal in Mirpur village, reports say.
The Times of India newspaper reported that that the villagers accused the school teachers of killing the students "in a fit of rage".
Poice said they were investigating the case, adding that there was no evidence the teachers were involved in the students' deaths.
Local TV channels showed footage of a group of men beating Mr Sinha with sticks and kicking him as he lay on the ground with his face down.
The school director later died from his injuries in hospital, police said.
The mob also set a couple of vehicles and two classrooms on fire.
Vigilante justice is not unheard of in India and there have been a number of such incidents in recent years.
In March, a suspected rapist was taken from prison by a mob and lynched in the north-eastern state of Nagaland.
Syed Sharif Khan, was stripped naked, beaten and dragged through Dimapur, Nagaland state's main city, before being hanged.
The director of a school in the eastern Indian state of Bihar has been killed in a mob lynching over the deaths of two of his students.