Passengers were evacuated from the twin-engine, wide-body jet to the runway using emergency slides, officials said.
According to preliminary information, the plane's left engine burst into flames on takeoff, said Ian Gregor, a spokesman for the Federal Aviation Administration's Pacific Division.
He said the plane was carrying 159 passengers and a crew of 13.
McCarran airport said on its Twitter feed that the blaze was extinguished and all passengers had been removed from the plane by 4:18 p.m. (2318 GMT), five minutes after the Clark County Fire Department and airport emergency crews received their first call of trouble.
A photograph posted by the airport of the immediate aftermath of the fire showed the plane idled on the runway with its fuselage charred.
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McCarran said its flight operations would continue on its three other runways, though the FAA ordered all inbound air traffic from several other airports held at their points of origin, including San Francisco, Phoenix, Denver, Salt Lake City and Albuquerque, New Mexico.
It was not immediately clear whether the two people McCarran said were taken to local hospitals were passengers or crew members.
Jacob Steinberg, a sports journalist for Britain's the Guardian newspaper, wrote on Twitter that he had fallen asleep on the plane during takeoff and felt the jetliner come to "crashing halt."
"Could smell and see smoke but was on other side of plane. One person said fire melted a couple of windows," Steinberg wrote. "They opened the back door and slide went down and smoke started coming in plane, followed by mad dash to front. A lot of panic."
Another passenger, Dominic Worthington, who lives in London, told NBC News in a phone interview that smoke could be seen on both sides of the plane, and then the evacuation chutes deployed.
"We were literally just about to take off," he said.
A spokeswoman for British Airways, which is owned by IAG, said safety is always the airline's priority and the airline is looking after customers.
Boeing Airplanes tweeted initially that it was aware of the incident and that its teams were gathering information, then later tweeting that Boeing was prepared to provide technical assistance to the National Transportation Safety Board.
- CNBC contributed to this report.