Poisonous plants are typically more dangerous when you burn them, at least that's true with plants that have oily toxins (poison ivy/oak). Toxins in plants aren't necessarily vaporized when burned. Smoke is a particulate, not a vapour. If you are burning something toxic, the toxins can potentially be carried by particles of smoke and be inhaled which is far more dangerous than being consumed in some cases. Even dead plants can still be toxic.
The toxin in Rhododendron and Mountain Lily is Grayanotoxin. Every material safety data sheet I've looked at says that Grayanotoxin is fatal if inhaled. But that's for pure manufactured Grayanotoxin II.
That being said, this guy thinks Rhododendron makes great firewood:
"Another property of rhododendron is that it makes excellent firewood –
even when it is green. It is a fast-growing hardwood with a very high
calorific content, and the stems of larger plants are so thick that
they can easily be sawn up into logs which are perfect for either an
open fire or a wood burning stove. I have taken rhododendron logs home
to burn in my woodburner during the winter, but a particularly good
use for the rhododendron firewood is on-site, where I need to keep
warm during the winter when I am there working. Burning rhododendron
means that I can generate firewood as I am clearing the rhododendron
from the woodland, meaning that I don’t have to have a big stack of it
seasoning and I don’t have to cut down broadleafed trees and season
them for use as firewood on-site, I can leave them standing and
growing. So rhododendron has become my staple firewood, leaving the
trees for better purposes."
And the Scottish government recommends burning Rhododendron to control its spread as an invasive species, so it would appear that it is at least less toxic than poison ivy or poison oak when burned.
They say that one of the first indications of Rhododendron poisoning is salivation, I suppose you could have a campfire in your yard and see if the smoke makes you drool... But I think the fact that the guy in the Blog is praising Rhododendron as fire wood, that he burns indoors and is still alive and blogging means that it's probably ok to burn for heat outdoors. I probably wouldn't inhale the smoke though... or roast hotdogs over a grayanotoxin fire...
Over 1000 species of rhododendrons/azaleas exist. All parts of the plant are considered poisonous, and as little as ingestion of 0.2% of an individual’s body weight can result in poisoning. When ingested, clinical signs include gastrointestinal signs (e.g., drooling, vomiting, diarrhea, abdominal pain, inappetance), cardiovascular (e.g., abnormal heart rate, heart arrhythmias, weakness, hypotension), and central nervous system signs (e.g., depression, tremors, transient blindness, seizures, coma, etc.). The overall prognosis is fair with treatment.
When burned the gryanotoxin is destroyed at temperatures of 150 degrees Celsius and above, and no evidence of toxicity has been found in the smoke or coals of the rhododendron plant. It is a hard long-burning wood and can be used safely.
However, inhalation of any smoke from any fire can have harmful health effects and should generally be avoided.