By RAW, the monster is not on fire. Burning Hands states (emphasis mine):
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.
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.