The short answer is that we are uncertain about the origins of the fire, but we can say with some certainty that Washington did no order the city to be torched during his retreat.

Washington's priority during the retreat following the Battle of Brooklyn Heights was to keep the army in one piece after being royally thrashed by the British regulars. The Continental Army had jut lost a fifth of its fighting force as casualties and Washington had only barely prevented the panic from immediately scattering the army. He knew that the army was in no condition to fight against the British, were preparing to lay siege to his encampment, and so he took efforts to escape without a fight.

His efforts were successful in this regard. The British awoke on the morning of August 30 to find the Continental Army had vanished right from under their noses. Washington had made a stealthy retreat to Manhattan with no casualties. He later engaged the British again at Harlem Heights on September 16, but his primary aim was to retreat from New York.

Washington had a knack for stealthy troop movement. He managed to stealthily set up ordinance on a hill overlooking Boston during his siege earlier in 1776 that forced the British to retreat. Notably, the British issued terms for this retreat that stated they would burn the city unless they were allowed to peacefully evacuate, to which Washington consented.

Washington and his soldiers and generals were fighting on their own territory, near their own homes. They had little desire to set fire to their homeland, though this was occasionally discussed in council. Washington also knew that many Americans were on the fence about whether or not to side with the Continental Congress or the Crown. Torching the fledgling nation's biggest and most commercially important city would have deeply demoralized the Patriots and the undecided.

A fire did break out, but it appears to have been accidental. Additionally, it occurred after the British had fully occupied the city. British General Howe suspected Patriot arsonists, but Patriots also assumed this to be a vindictive act on the part of the British Army.

Sources:

Ron Chernow - Washington: A Life

David McCullough - 1776