Alitalia said in a statement that all airlines have been forced to reduce flights.
Italy’s flagship carrier added that all of its passengers, as well as those flying with its merger partner, Etihad, will have to check in at terminal one, regardless of their destination.
Passengers will be able to change tickets for flights that were due to depart on Friday, or get a refund.
The company said it would re-route as many passengers as possible, as well as deploy larger jets to offset some of the disruption, but advised people to check their flight status four hours in advance.
The city's main airport was closed to passengers for several hours on Thursday after a blaze triggered by an electrical fault in a bar broke out in terminal three, which is used for international flights.
When the airport reopened, there were scenes of total chaos with thousands of people trying to find out what was happening and frantic airport staff desperately attempting to deal with the backlog and match disorientated and frustrated passengers to seats on waiting planes.
Three airport employees were treated for smoke inhalation but there were no serious injuries.
The fire erupted shortly after midnight and was not brought fully under control until more than five hours later, by which time the authorities had decided to cancel most of the morning's flights and close the airport to passengers.
A shopping area in the terminal which is home to a string of upmarket boutiques was devastated.
"The bit beyond the security gates where the shops are has been completely destroyed but the rest is usable," said Lorenzo Lo Presti, the head of Airports of Rome, the private company which runs Fiumicino.
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The chaotic scenes prompted calls for a probe into why a fire limited to one terminal had caused such disruption over the entire airport for hours.
— Alessandro Antinelli (@AleAntinelli) May 7, 2015
"You have to ask why the company that runs the airport did not have adequate plans in place to deal with an event of the kind that happened overnight," said Michele Anzaldi, a member of the National Assembly.
"If it had to resort to closing everything it means something has gone wrong," the centre-left deputy added. "There was no plan B."
Anzaldi and other lawmakers said the day's events showed Rome was far from ready to cope with an anticipated surge in tourist arrivals as a result of the jubilee year declared by Pope Francis from December 8th.
Firefighters said it could take several days to make the area hit by the fire safe.
The terminal is one of four at the airport and serves flights to European countries outside the Schengen no-borders zone and destinations further afield with the exception of Israel and the United States.
Vito Riggio, president of the national civil aviation authority, told reporters at the airport that the cause was "probably a short circuit".
Police do not suspect any foul play.