That is not how the property tax system works. I used to work for a tax assessor office, and tax revenue is based on (total assessed value) times (millage rate). If the total assessed value goes up, either with new construction, improvements, or inflationary pressures, then the millage rate could simultaneously go down, as mandated by law in some states, to keep the same total revenue.
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.)
I realize property is taxed that way. My mom lives in Maryland. They are required to publish the constant-yield rate, which is the millage level necessary to yield the same amount of dollars. Yet each year they continue to tax at the same rate - an effective tax increase - but tell their constituents they have kept taxes the same. The assessed value can be challenged every year and that value will be based on comps of recent sales.