The London Fire Brigade said about 70 firefighters were tackling the blaze in the west of London, which caused a mass power outage at Heathrow, Europe's busiest and the world's fifth-busiest airport.
Huge orange flames and smoke could be seen shooting into the sky.
About 150 people were evacuated from nearby buildings and thousands of properties were without power.
The fire brigade said the cause of the fire was not known.
"To maintain the safety of our passengers and colleagues, Heathrow will be closed until 23h59 on 21 March," Heathrow Airport said in a post on X, adding that passengers were advised not to travel to the airport.
According to flight tracking website FlightRadar24, at least 120 inbound flights to Heathrow were having to divert to other airports.
It said at least 1351 flights to and from Heathrow would be affected on Friday, not including flights that might be cancelled or delayed due to aircraft being out of position.
British Airways itself had 341 flights scheduled to land at Heathrow on Friday.
"Heathrow is one of the major hubs of the world," said Ian Petchenik, spokesman for FlightRadar24.
"This is going to disrupt airlines' operations around the world."
The impact was immediate. Qantas Airways sent its flight from Perth to Paris, a United Airlines New York flight headed to Shannon, Ireland and a United Airlines flight from San Francisco was due to land in Washington, DC, rather than London.
Some flights from the US were turning around mid-air and returning to their point of departure.
Travel experts said the disruption would extend far beyond Heathrow.
Airlines' carefully choreographed networks depend on airplanes and crews being in specific locations at specific times.
Dozens of air carriers will have to hurriedly reconfigure their networks to move planes and crews around.
"The other question is, 'What will airlines do to deal with the backlog of passengers?'", said travel industry analyst Henry Harteveldt with Atmosphere Research Group.
"It's going to be a chaotic couple of days."
A Heathrow spokesperson told Reuters in an email that there was no clarity on when power would be restored, and they expected significant disruption over the coming days.
On the ground in London, a number of homes and businesses were without power.
"Firefighters have led 29 people to safety from neighbouring properties, and as a precaution, a 200-metre cordon has been established, with around 150 people evacuated," the fire brigade said.
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