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?