A PC in my campaign uses burning hands all the time. He keeps asking, "If the monster is on fire, does the monster keep on taking damage automatically?" Does it?
The monster is on fire because it was wearing clothes and leather armour which presumably become on fire after burning hands.
By RAW, the monster is not on fire. Burning Hands states (emphasis mine):
Burning Hands:
The fire ignites any flammable Objects in the area that aren't being worn or carried.
roll20 compendium and PHB p.221
Thus, you could use Burning Hands to start a campfire, brush fire, or even light a barn on fire. But not set a creature or its clothing on fire.
Related topic:
Spells only do what they say they do, so if continued damage was intended then the spell would say so.
There is no general rule for being on fire. Effects that can cause a creature to be on fire will specify how much damage is done for how long. Different effects will cause different amounts of damage.
