The incident occurred at Camp Pendleton, California while a Marine battalion was undergoing a combat readiness evaluation, Marine spokesman Paul Gainey confirmed.
"Officials are investigating the circumstances surrounding the incident at this time," his statement said.
No other details were given about the fire or possible causes.
The injured Marines were evacuated to hospitals in the area, with eight taken to the Burn Center at University of California San Diego Health.
Three of those were in critical condition and five in serious condition. Two more were in critical condition at the University of California Irvine Medical Center, the Marines said.
Amphibious Assault Vehicles have been used by the Marines since the 1970s and are designed for landings from the sea.
Wednesday's incident followed two other fatalities at Camp Pendleton, the Marines' biggest base on the west coast of the United States.
On August 30, a 22-year-old Marine was found dead during a training exercise. The cause of death has not yet been established.
In 2015, a 19-year-old Marine was found shot dead in the head on a firing range at the base.
In mid-June, seven sailors were killed when the destroyer USS Fitzgerald collided with a container ship off Japan.