In the subtitles of the Japanese version, we learn that the Kohaku River is still flowing underground, so it hasn't technically been "destroyed", although the English dub just says that they covered up the river and now an apartment building is over it. But what would happen if it was completely destroyed and filled up? Would Haku die? We saw that spirits' deaths are possible, such as when Haku was going to die for stealing Zeniba's gold seal before Chihiro unknowingly removed the curse on it out of love.
So for example, if radishes became extinct, would the Radish Spirit die? If a specific type of white fox (the Byakko) became extinct, would Lin (the spirit of Byakko), die?
That would probably be the case. Or they could perhaps become a creature like No-Face, lost and formless. Certainly, I think if a spirit is tied to one specific thing (like one particular river, one tree) and that gets destroyed, they would die. They lose their existence in the human world.
The collective spirits are harder, like the radishes, since that physical existence is more numerous, more like a concept. Spirits and gods are tied to belief, so if all existence and memory of something disappears, the spirit loses the source of its life. At the very least, I think no more of those spirits could be born, like Lin would only exist in the spirit world and eventually die of old age/some other cause.