The case is being brought against transport secretary Chris Grayling after plans for the airport's expansion were approved by Parliament last year.
Objectors, including environmental groups Greenpeace, Friends of the Earth and Plan B, say the airports national policy statement (NPS), in which the government set out its support for a third runway, failed to deal adequately with the impact of the project on air quality, climate change, noise and traffic congestion.
If they succeed in persuading two senior judges that the NPS is unlawful and should be overturned, the expansion plans would have to be put to another parliamentary vote, extending yet again the marathon planning and environmental process of considering Heathrow's future.
Grayling says the new runway will set a "clear path to our future as a global nation in the post-Brexit world" and the proposal won an overwhelming majority of 296 in a Commons vote.
A construction start date of 2021 is planned, with the third runway entering service in 2026.
The hearing at the Royal Courts of Justice, before Lord Justice Hickinbottom and Mr Justice Holgate, is expected to last two weeks, according to the court's agreed timetable.
Laura MacKenzie, of Friends of the Earth, said before the case: "An expanded Heathrow Airport would put seven hundred extra planes a day into our skies, pumping millions of tonnes of carbon into our atmosphere.
"With the impacts of climate chaos already being suffered by millions around the world, we simply cannot allow this to go ahead."
Tim Crosland, director of Plan B, said: "We are enveloped in a climate emergency, demanding an urgent and radical change of course.
John Sauven, executive director of Greenpeace UK, said: "Governments are very happy to talk the talk when it comes to protecting the air we breathe and the climate we all share but, unfortunately, getting them to walk the walk often takes legal action."
The Department of Transport has said that it anticipates legal challenges to major infrastructure projects but will robustly defend the decision to allow the Heathrow expansion.
DfT officials say that the High Court case will not impact on the on-going planning process or the timetable for delivery of a third runway.
Last week, a High Court judge ruled that a consultation on a policy on fracking in the National Planning Policy Framework was unfair and unlawful and the government had failed to take into account up-to-date scientific evidence on the climate change impacts of such development.
R on the Application of London Borough of Hillingdon v Secretary of State for Transport. CO/3089/2018