One of my players seems to think that this spell will cause all sorts of carnage whereas I'm more inclined to say it may cause a bit of a fire and a bit of damage but not necessarily be all-consuming.
Are there any rulings on this?
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The description of Burning Hands states that it 'sets fire to flammable objects'. At its maximum, it is a 15-foot wide cone. So the answer here is that it might be able to set fire to a wooden building, but wouldn't necessarily burn it down, unless the fire was ignored for a reasonable length of time.
Remember, too, that even in medieval times there were still some measures that could be taken to prevent house fires, and that wood doesn't just burst into flames at the touch of fire. Some varieties of wood are particularly difficult to ignite, and if it's rained recently it probably won't matter at all, it's just not going to light up.
To set fire to a building, you would either need to coat it in something more flammable, such as alcohol or pitch, or keep a fire on it for an extended period of time. A momentary blast of flame might singe or char a wall, but it probably won't ignite it. Think about it: building a campfire, you have to start with kindling and small twigs, you can't expect logs to burn straight away — same here.
Oddly enough, it would probably be easier to use a torch for your arson, since you could hold the torch to the wood until it ignited. Alternatively, the 'flask of oil' item is pretty much made for this; its whole purpose is to burn easily. If you throw a flask of oil on the building and then cast Burning Hands, you should achieve your goal. (Assuming your player wanted to burn down the building, that is; if he was worried about property damage when casting Burning Hands he should be good to go.)
Depends where you aim it.
Contrary to what you might think, wooden walls (especially ones used for building things) are not super flammable. Non-treated wood has a flash point of 300 C, which is about 80 degrees hotter than what it takes to ignite paper. This means that it takes a very long, or very hot fire in order to make wood light on fire. Think of it this way: if a wall was flammable, then just having a torch in a sconce on the wall would eventually light the wall on fire. Since that doesn't really happen, we can assume that walls are at least moderately resistant to fire.
If you cast burning hands at a wall, it'll certainly scorch and might throw off a few sparks. It's unlikely to go up in a sheet of flame.
However: Roofing, especially pseudo-middle-ages roofing, is likely not made of flame-retardant materials. Thatch lights better than kindling, and tar is reasonably flammable as well. For comparison: the flash point of paper is about 230 C, the flash point of tar is between 100 and 300, depending on it's makeup. If you aim burning hands at the roof, then the building will be very much on fire, and probably quickly.
As of the current date, we don't have solid rules for damaging objects in 5th edition, since the DMG isn't out yet. We can look at previous editions for a loose guide, though. In 3.5, wood had a hardness of 5, and likely took half damage from fire, since it's an object. A 5th level burning hands did an average of 12.5 damage. 12.5/2 - 5 = 1.25 damage per attack. This means that it would take 8 spells per inch of thickness to burn through wood, which strongly implies that it isn't going to be affected overmuch by flame.
There are some factors that play into this spell not really doing much to the construction types of typical small inns.
The use of gypsum plaster to coat walls (both interior and exterior) was commonplace even in antiquity (going back to far before the Roman era), and thick plaster coats behave similarly to sheetrock (drywall) in resisting fire, at least for a period of time. There are some test results performed on lath-and-plaster ceilings; walls, especially ones where the plaster was laid directly over the structural surface as opposed to standing off by way of furring and lath, will likely perform somewhat better than the numbers given here.
In addition, heavy log walls, or even rough-hewn timber beams supporting other wall and roof types, behave much differently from the light-frame wooden walls and roof trusses we are familiar with from this day and age. In fact, this distinction is captured in modern fire codes: heavy timber structures (this is defined by code to be construction where all members are a minimum of 8" or 20cm in diameter) are considered Type IV (mill construction), which is rated for an hour of fire resistance; typical residential light-frame construction (Type V-B) has practically no fire-resistive value to it, however. See some construction type definitions for a quick rundown on how fire codes define the various construction types, and this piece specific to log walls (read from page 12 on, and in particular the test results cited on page 19) for how 'pure' log construction relates to its mill brethren.
So: you may mildly singe, or at best scorch, exposed log walls, and would likely do little to nothing to a well-plastered wall. Roof thatch, though, would catch alight more easily -- albeit not as readily as the mental images others create. However, the thatch and laths would fail well before their support beams (assuming heavy timber supports), as once thatched roofs do catch alight, the resultant fire will spread relatively swiftly, leading to total collapse of the thatch and lath. See this, this, this, and this for details.