What is it with saying "asking for a friend" that makes people know that you're asking for yourself but are trying to hide it for some reason? I'm asking for some friends, so they can ask questions for me or something the details are not important.
Strictly speaking, a typical fire damage breath weapon doesn't set things on fire. It just deals fire damage, and unlike fireball, it lacks the little bits about setting flammable material alight. There's nothing in the rules, therefore, that explicitly allows this. Nothing that denies it either, though. Specific breath weapons may have their own rules on the matter, but they usually just follow the standard wording if they don't just refer you to the general breath weapon rules.
It's the kind of thing that makes sense, though, from a simulationist perspective, so ask your DM's opinion on the matter. Personally, I'd be inclined to rule in favor if I were DMing... except your you, Sirpercival. If I were DMing for you, I'd rule against it, hard, and probably dock you a few thousand xp for even asking such a silly question. Good thing you're asking for a friend, right?