Should I continue to pay the property taxes if I am not paying my mortgage?

In California the property taxes are a super lien, which means that the lender is junior to the tax collector. A lender will pay past-due taxes to secure their lien position. Once the property goes to foreclosure the lender will have to pay all past due property taxes. If foreclosure is eminent, there is no reason for you to pay the property taxes. That said, if you live in a state that allows deficiency judgments, this amount would be added to the deficiency, so it may be cheaper to pay them before penalties and late fees are added.