The protesters announced last month they were intending to stage several days of non-violent action over plans to expand the airport with a third runway.
It was due to start on Tuesday, with further demonstrations planned for ten days from July 1, and threatened to cause chaos at a peak time for holidaymakers.
A leaked internal memo had shown drones would potentially be used to stage the protest , prompting the government to warn that any such demonstrations would be met with the "full force of the law".
Heathrow also said using drones would be "reckless" and could endanger lives.
But the campaign group announced today that it had decided to cancel the action.
It said in a statement: "Extinction Rebellion (XR) will not be carrying out any actions at Heathrow Airport in June or July this year, aimed at causing disruption to holidaymakers and those planning to use the airport in this period.
"The Heathrow Airport authorities will therefore not have to pause any summer flights."
However, the statement also included an "action plan" which the group said could be followed if the plan is revived.
It said Heathrow had not been removed from its strategic plan and that the government's plans for a third runway "could not be more incompatible with the imperative to reduce greenhouse gas emissions globally".
The Metropolitan Police had issued a statement on Friday warning that any protestors who disrupted the airport would be dealt with in line with the law.
It said the planned action had the potential to cause "widespread disruption" and urged activists not to use drones in an effort to suspend flights.
In a plea to protesters, Deputy Assistant Commissioner Laurence Taylor said: "As we have seen, the illegal use of drones at airports also has the capability to cause great disruption to the public, not only in London, but internationally.
"While it is our duty to deliver an appropriate plan for this protest, I want to make clear the impact this kind of activity has on frontline policing.
"This has the potential to cause widespread disruption, and in order to prevent this happening we have drawn vital police resources from across the capital."
He added: "We would urge anybody intending to join this event with a view to committing criminal activity, whether considered peaceful or not, to strongly reconsider."
The group, which had previously insisted it was still deciding whether to use drones in the protest, added in the the statement that accusations that XR was willing to endanger life was a "depressing and predictable smear".
XR said if drones are used for future demonstrations that operators would fly them at a maximum height of six feet within the restricted 5km zone surrounding Heathrow.
They would also not be flown within flight paths, but could still be used in areas that could force the airport "to safely close airspace", the group said.
The group added that airport authorities and the public would be given two months' notice of any planned action.
In December, there was chaos at Gatwick Airport for three days, when drone sightings resulted in around 1,000 flights being grounded, affecting 140,000 passengers.
The Ministry of Defence introduced detection systems capable of being deployed across the UK to combat the threat of drones in the wake of the Gatwick fiasco.
Extinction Rebellion has carried out several separate demonstrations over the past year.
In the most recent one on Friday, Extinction Rebellion Lewisham "swarmed" roads in south-east London in an attempt to put pressure on the Government over air quality .
Air pollution in the borough of Lewisham was found to be approximately six times higher than safe levels recommended by the World Health Organisation in 2017.