A dragon that breathes fire breathes real fire
If you observe the ability of the red dragons' breath weapons, for example, the fire-breathing action of the wyrmling red dragon says:
The dragon exhales fire in a 15-foot cone.
Fire and flame are synonymous, therefore, if you had some kind of effect that would extinguish the flame continuously, the fire breath of a dragon would also go out.
The breath of a dragon is not magical
According to the Compendium of Wise Councils, what does this have to say in "Is it the breath of a magical dragon?":
Determining if a feature of the game is magical is simple. Ask yourself these questions about the feature:
- Is it a magical object?
- Is it a spell? Or does it allow you to create the effects of a spell?
What is mentioned in your description?
- Is it a spell attack?
- Is it powered by the use of spell slots?
- Does your description say it's magical?
If your answer to any of those questions is yes, the characteristic is magical.
Since the dragon's breath weapon does not meet any of the guidelines, it is not magic.
Therefore, since the breath weapon creates a non-magical flame, there is no reason why this object does not extinguish it.