Do I zero out property taxes paid for state if my owned home is in another state, and I am military?

It depends.  First I assume you are asking about a specific state return.  Perhaps the question relates to a property tax refund for that state.  If that is the case and you did not pay property taxes to the state in which you are filing then, yes, you would leave it blank or put in zero as required.

You mention you are military so I assume you own a home in the state that you claim as your home of record which is not the state you are currently stationed in.  If that is the case unless you earned income (non-military source) in the state where you are stationed or that state withheld taxes from your military pay, then you probably do not need to file a return in that state.  Typically, you only need to file in the state that is your home of record, which would be your resident state for tax purposes.  

Click here for more information on filing state taxes for active military.