My guess is dragons breath is produced by a combination of two glands that produce chemicals that are hypergolic. This is seen in nature in bombardier beetles (which use hydroquinone + hydrogen peroxide).
The smell would certainly depend on the combination of chemicals, but a hot burnt and charred smells are probably common to all possibilities. The unburnt chemicals are likely to be irritants, being either strong oxidizers or reducers.
Overall, even after the main blaze has cooled, I would expect the area (and downwind) to be unsafe for humans and respiratory issues for those exposed to the remnants.