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.
In a statement, 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 on board 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," Steinberg writes.