If the Nazgûl are neither dead or alive then would it really matter to them if they were set on fire? I'm talking of when Aragorn fights the Nazgûl in the film when Frodo is stabbed.
They do not feel pain as we might, they have no bodies. The power of light is a strong theme in the Tolkien mythos. The Nazgûl are not afraid of the fire as a thing that might harm them from its flame, but because of its light.
The Nazgûl are creatures of fell darkness, sustained by the magic of their Dark Lord. At their weakest, they expected little resistance. But not only did they find resistance, they were confronted by ancient magical weapons, the very kind that can harm their immaterial bodies, and light boldly presented. The did the only thing they could. They fled.
The Nazgûl are not alive, in fact, they don't really have bodies to speak of. They are wraiths, spirits capable of interacting with the world. They are powered by and filled with darkness and they sustain themselves on the fear of their enemies.
Light, any light, even as mundane as fire can affect them early in the series. The Nazgûl were unable to operate in daylight at this point. Sauron's power was weak early in the series, and the Nazgûl were counting on their ability and legend to inspire fear and helplessness in their enemies. They were also weak, relatively speaking. At Weathertop they are only formidable in a group.
Note that their power grew as Sauron's did. By the battle of Gondor, the Witch King is seen in daylight, rides a fell beast and laying waste to the enemy in battle with his giant flail, his screams causing fear and terror among the combatants. Imagine how powerful they may have become if Sauron had been successful.