A coalition of local councils, environmentalists and London residents claim the government has failed to properly address the impact on air quality, climate change, noise and congestion that expansion would bring.
London Mayor Sadiq Khan is also backing the lawsuit.
Demonstrators gathered outside the High Court on Monday for the first day of a two-week hearing.
Parliament approved plans last year for the third runway, backing what the government described as the most important transportation project in a generation.
Prime Minister Theresa May has said the expansion will boost economic growth.