Is It Possible for Property Taxes to Be Too Low?

In Louisiana, the answer is probably "yes."
An excellent editorial in the Daily Advocate notes that the ten lowest-property-tax counties in America in 2006-2008 (for residential property) were all in Louisiana. The main reason for this is pretty straightforward: every owner-occupied home in Louisiana receives an exemption for the first $75,000 of home value. In practice, this amount covers something close to half of all properties in the state.
The topic is being raised right now because there are active proposals to increase the homestead exemption. As the Advocate editorial correctly notes, the impact of this would be a

shift of tax burdens directly onto owners of business and commercial property. It also would shift more of the property tax burden onto renters, rather than homeowners; renters pay property taxes through their monthly checks to landlords, without benefit of a homestead exemption.

This isn't an argument that Louisiana's homestead exemption actually needs to be reduced. But it certainly makes a good case that further increases ought to be the last thing on state policymakers' minds at this time.