Well, there are established fictional descriptions, or indications.

The one that I think would hold up the best would be the way Pern (Anne McCaffrey series) handled it. In that series, the dragons weren't your typical fantasy dragons, they had been genetically engineered from a smaller creature called "fire lizards". So there was an attempt to lean harder into sci-fi than fantasy from the beginning of the series, even though it didn't seem like it.

That's kinda tangential, but McCaffrey did put some thought into how they shot fire.

The dragons chewed up phosphorus bearing rocks, swallowed them into a special stomach, and the results of that breaking down allowed them to basically belch flames.

If you're going to ignore "magic" as the source of the breath weapon, then that method makes sense. Since phosphorus sulfide could conceivably be made inside some kind of complicated organ like that, as could other phosphorus based compounds, the obvious choice is for dragon breath to smell like matches.

Now, that assumes you meant what the breath smells like because they breathe fire. There are other versions of dragons that shoot other things. But, more importantly, shooting fire isn't actually the same thing as breath. It's a separate process from breathing.

So if a dragon was in front of you, breathing in and out of its mouth, it wouldn't be ONLY the smell of the fire starting agents. You'd smell the brew of saliva, any leftover bits of food left in the mouth, and likely some degree of "funk" from whatever extra compounds were in the saliva to help it protect the dragon from its own fire.

Imagine a snake's breath, but spicy ;)