Airport officials said at least 14 people were taken to hospital with minor injuries.
US Federal Aviation Administration spokesman Ian Gregor said the left engine of the Boeing 777-200 caught fire before take-off.
The plane was seen engulfed in flames and smoke. The fire was later put out.
British Airways said the aircraft "experienced a technical issue as it was preparing for take-off".
In a distress call the plane's captain asked for the airport's emergency services: "Speedbird Mayday Mayday. Speedbird 2276 request fire services."
"Speedbird 2276 heavy, we are evacuating on the runway. We have a fire, repeat, we are evacuating," he adds.
The pilot later told passengers that the accident was the result of a "catastrophic failure of the engine", reports the Guardian's Jacob Steinberg, who was onboard the plane.
"We looked out the window and for the first time saw smoke near the wing. We could smell it. It was like burning rubber - bitter and deeply unpleasant - and it was time to panic,"
At the scene: Lynne Alexander, passenger
It all happened so fast and the crew were amazing.
The plane was jetting across the runway to take off, when there was a bump and the plane starting screeching to a halt.
I was sitting on the right hand side near the wing, but I could see the smoke from the left engine belching out. Debris was falling from the sky and I could smell smoke and fuel so knew it was a fire.
People were getting out of their seats and the crew were trying to calm them down.
Then the pilot said, "emergency evacuation... off plane now". People were pushing and shoving and screaming. There was panic but then the chute was deployed and down I went. Then we saw the flames and smoke.
We all ran far from the plane as the fire crew arrived and doused the engine.
Close call, but we're alive. BBC