Yes, bullets can fire without guns because the firing is a reaction to heat, the primer being triggered, or simply enough momentum to propel it. A firearm is only one way to fire a bullet as it is designed specifically for striking the primer and using the barrel to build velocity from the force which is a result of the explosion inside.
The chances a bullet will fire from being dropped are minimal, but there are other ways to fire the cartridge.
A non-explosive way would be a slingshot or non-explosive propellant like an airgun. There is no explosion and the velocity will be minimal, so it’s unlikely to penetrate deep and have the same effect as a gun, but it is technically in flight and can do damage. This is more like in the movies when you see a firearm hidden in a book or in a box and it’s silent because it’s propelled more like a cross bow than a firearm, or like a dart from a blowgun.
Although it may not fly, or at least not in a linear and curved direction, heat can cause what is known as “cook off” which is when the ammunition combusts and causes the bullet to fire or explode. Cook off was a concern when machine gun style firearms first came out because the cooling systems were not as advanced. That was until the Maxim Machine Gun was invented, which is part of the story of how silencers were invented.
According to wikipedia, some types of rounds with smokeless powder can cook off at 160-170 degrees centigrade, while a study on Science Direct shows another type was igniting at 151.4 to 153.4 degrees centigrade. So good news, chances are your ammo won’t cook off as these are extreme temperatures.
With that said, you may be asking about fire proof gun safes. At these temperatures your firearms would already be damaged and it would be too late to do anything. It’s one of the reasons that fireproof ratings on gun safes no longer make sense.
The third and final situation a bullet may discharge without being in a gun is when the primer is struck with a lot of force. This too is unlikely because it requires precision, accuracy, and for a direct hit on a small area. It has to be a perfect case scenario, and this applies to dropping bullets accidentally.
Cartridges are designed aerodynamically to fly point or tip first by reducing or directing the air around them. This causes them to fly or fall tip first. The primer which would cause the cartridge to ignite and fire is on the back of the cartridge and likely would be the first point of contact on the ground or target.
The chances of ammunition firing without a gun are slim to none. Yes, it can happen, but it is unlikely unless it is being done on purpose.