Local and state governments do. They derive a significant portion of their revenue from the property tax. Their bonding authority is tied to a reliable revenue stream from property taxes year in, and year out. If housing prices have to come back to reality, then property tax revenues evaporate. Local school funding dries up, and government has to raise taxes elsewhere.
Are you kidding? If you want higher state and local taxes just tell your local elected to raise them. Government Propping up house prices artificially nationally (using printed money yet) hurts home buyers and is a bad way for you to keep your local tax rates high.
If the total assessed value goes down, due to depreciation formulas, knockdowns, or loss of comparative sales value in a declining economy, then the millage rate could go up to keep the same revenue.
Also, foreclosures should have absolutely no effect on property tax revenues, since the bank that owns the property would be responsible for making sure the taxes are paid, so that the property does not get sold in a tax sale to someone else (who would then pay back taxes plus interest, and then be responsible for property taxes thereafter.)