Heathrow Pause - a splinter of the Extinction Rebellion movement - said it intends to fly the machines in the 5km exclusion zone around the transport hub on Friday.
But protesters could find themselves facing life in jail if they are found to be endangering the lives of passengers, police have warned.
Met Police Deputy Assistant Commissioner Laurence Taylor said: "We will be arresting anybody who commits unlawful offences.
"We are really clear that it is unlawful, it is a criminal offence, and anybody who turns up expecting to fly drones in that exclusion zone will be arrested."
Frontline officers have been drafted in from other areas of the Met to tackle the protest.
Mr Taylor said: "We can't compromise the safety of passengers.
"If it does turn out to only be a handful of people, they will be responsible for the fact we've had to put that policing plan in place."
A meeting was held between Heathrow Pause activists and the police force earlier this week to discuss Friday's action.
The Met has reassured passengers that it is working hard to minimise any potential disruption.
Mr Taylor said: "I'm really confident in the plans we've got in place, the level of resource and the tactics available to us, and we obviously take passenger safety incredibly seriously.
"I would urge people to come on Friday because we're going to do everything we can to prevent any disruption and I am confident we will be successful in that."
Climate change activists have held a number of disruptive protests this year.
Most notable was the shutdown of a number of sites in central London in April by Extinction Rebellion, including Oxford Circus and Waterloo Bridge.
Mr Taylor urged the protesters to consider their impact on the public.
"Having to arrest hundreds of people is a huge demand, not just on the police, but on the justice system. It's very costly.
"We would urge protesters to consider that wider and considerable impact.
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"People absolutely have the right to protest, but that can't be at the expense of thousands of other people."
Heathrow Pause said it has "no choice" but to carry out the action and invited the Prime Minister to join the protest.
It said: "The current Prime Minister is on record for saying his opposition to Heathrow expansion is so profound that he would 'lie down in front of the bulldozers'.
"We invite Boris Johnson to join us in flying a drone and showing his commitment to stopping Heathrow expansion and addressing the climate and ecological emergency."