The move is being made in response to the summer cap on daily passenger numbers imposed by the UK’s busiest airport.
BA’s flights to domestic and European destinations will be affected by the sales halt.
Thousands of air passengers have been inconvenienced in recent weeks, including last-minute cancellations.
Airports and airlines that cut jobs during Covid lockdowns have struggled to find enough staff as holiday demand has increased.
Heathrow Airport has struggled to keep up with the rebound in air travel, and issues with its baggage handling systems have resulted in massive delays in reclaiming luggage for passengers.
Heathrow told airlines last month to stop selling summer tickets because it planned to limit the number of passengers departing each day to 100,000 – 4,000 fewer than previously planned.
The number cap is set to remain in effect until September 11th.