First, presume that the players aren't the first people in the world to ever think of obvious uses/abuses of magic. They aren't dealing with a bunch of mundanes who've no idea if/how/that Magic works.
That's a good thought to keep in mind, not just in this situation but whenever when players try to get too creative.
Next I'd like to observe the difference between real world operations and game mechanics: In the real world that very solid door would be made of solid timbers, probably oak, and that don't light easily. Further, if they're in a damp setting (such as any underground area) they'll flicker out fairly quickly. Think of the Firebolt as being roughly the equivalent of a cigarette lighter.
Lighting a campfire usually requires tinder and kindling, small,. easily lit pieces of dry material. Even with a lighter you don't jus light a log on fire.
That's the real world. Can you envision trying to light that door, and burn it down, using a lighter? Can you see that image?
Good. Now wipe it from your memory, because that has nothing to do with game mechanics.
Game rules don't ever deal with smoke from burning doors, or the fact that these will consume the oxygen in a tight, underground space. Nobody worries about carbon-monoxide poisoning, and wood is flameble, whether it's a handful of matchwood or an ancient and damp timber from the hold of a ship. The rules make no distinction.
So, by the book, the trick works.
Now, what could keep it from working?
Presume that the timbers are soaked in creasote or some similar preservative. Also figure that the people who built them, as part of a defense, were aware of that cantrip, among other things.
Now actual creasote comes from a plant found in the "New World". Like corn, potatoes and tobacco (i.e. "pipe weed"), none of these existed in pre-Columbian Europe, the world so many of our games are modeled after. So don't worry about how creasote actually burns, that isn't what the doors were treated with.
So what was that door treated with? Something fire retardant? Something that puts off a truly noxious smoke when burned? As in, Poison saves for breathing in the vapors? Big hazard in case of a real fire, but stone buildings just don't burn all that easily or often, and it's not like there were actual fire codes to contend with.
So booby trap the doors.
Think along those lines. Also, think about traps designed to go off specifically when the doors are burned. Some small Alchemical thing that will burst when heated, spattering flaming liquid at everything within five feet. An ampule or three embedded right into the wood would be simple and relatively inexpensive.