The airport's demands are "entirely unreasonable and unacceptable" according to the airline.
Airlines will have to cancel more flights because of a daily cap on departing passenger numbers that was set on Tuesday.
The number of passengers at the airport will be reduced to 100,000 a day in order to deal with ongoing disruptions.
In a statement, the airline said it was regrettable that the airport gave the airline 36 hours to comply with capacity reductions.
The statement said that their communications dictated the specific flights on which we should throw out paying passengers.
We reject these demands.
dnata, part of the Emirates Group, is fully ready and capable of handling our flights, according to the company.
Since October 2021, the A380 flights have been reintroduced by the company.
They want to force Emirates to deny seats to tens of thousands of travellers who have paid for and booked months in advance, their long-awaited package holidays or trips to see their loved ones.
At a time when many people are desperate to travel after two years of Pandemic restrictions, this is during the peak season.
Mirates believes in doing the right thing. With all flights running full for the next weeks, re-booking the potentially impacted passengers is impossible.
It will be difficult to find new onward connections for 70% of our customers from LHR because they are headed out of the country to see loved ones.
It's not realistic to move passenger operations to other UK airports at such short notice.
It's not easy to find a parking spot at a mall for a widebody long-haul aircraft with 500 passengers.
The airline plans to operate as usual until further notice.
Travel Weekly asked for a response from the airport.
The picture is fromshutterstock.com.