It was unclear why the firetruck entered the runway while the plane was taking off.
This is the second incident in less than a month involving LATAM Airlines, after one of its planes had its nose destroyed during a severe storm that forced it to make an emergency landing.
Chile-based LATAM Airlines' Peru branch said it was deploying all its resources to attend to those affected, and work with authorities to support the investigation.
Lima Airport Partners, which operates Jorge Chavez, Peru's most important airport, said the airport will remain closed at least through 1 p.m. (1800 GMT) on Saturday.
The flight was LA2213, covering the domestic Lima-Juliaca route, LATAM Airlines said.