The incident occurred at about 2000 local time (0300 GMT) on Friday, said a Phoenix fire official, Capt Shelly Jamison.
About 15 people received minor scrapes and bruises, and one person was taken to hospital with shoulder pain.
Confirming the safe exit of all 298 passengers and 18 crew aboard Flight BA288, British Airways said the aircraft had been evacuated "following the usual procedures".
Passengers used slides to leave the aircraft
Charles Woolf, 16, was sitting at the rear of the plane, waiting to return home to Fareham in the UK after a holiday.
Everything appeared normal as the plane taxied out, he told BBC News, but after it had been waiting about 10 minutes for take-off, "a strange but harsh smell filled the cabin - it made my nose tingle and burned the back of my throat".
"Some people started to cough, and a few started to worry and panic," he said.
"After about five minutes, more people started to cough, someone towards the front of the aircraft shouted 'fire', the doors of the aircraft opened and the slides were deployed.
A passenger filmed the scene after being evacuated down emergency slides
"People were pushing a bit to try and get to the exit, but I'd say within 20 seconds of the doors opening I was out of the plane.
"I never realised how fast you can go down those emergency slides - a lot of people, including myself, have little friction burns on their elbows."
The 16-year-old praised the cabin crew's professionalism.
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