Fill a paper cup with water and place above a flame.  As long as the cup still has water, it will not burn.  The paper nearest the flame maintains the same temperature as the boiling water.  Since temperature is constant during a phase change, the water never gets above 100°C.  Due to conduction, the cup and water seek thermal equilibrium.  The temperature of the cup doesn’t raise high enough to ignite the cup.  If you do not have a flat bottomed cup, the edges of the cup that hang below the bottom will get scorched.

Paper Cup That Doesn't Burn