The UK's largest airport asked airlines to cut the flights because it was expecting more passenger numbers than it can currently cope with.
Some passengers did not know that their flights were cancelled until they arrived at the airport.
Heathrow said the cancellations were necessary for safety.
Travel writer and broadcaster Andy Mossack tweeted that there was "total chaos" and "zero customer service" at the airport on Thursday morning.
Mr Mossack, who was due to fly to Geneva at 08:25am, told the BBC he was told of the cancellation via an email about 6.00am which he didn't see until he arrived at the airport.
"There are no flights to Geneva until Sunday or Monday. So I've had to come home".
Terminal 5 was "awful... there were hundreds of people there. Some sleeping on the floor".
Another passenger tweeted that terminal 5 was a "disgraceful shambles" after he arrived on Thursday morning to find his flight cancelled.
PA reported that one passenger, Andrew Douglas, said he had spent four hours in queues to find out at check-in that his flight had been cancelled with no prior notifications.
Other travellers complained of poor customer service and a lack of help when trying to rebook their flights.