You only file in Virginia if you have Virginia-sourced income. That would include a job, lottery prizes, and real estate income such as if you rent the home or when you sell it. But just sitting there, no, you don't file a Virginia return. (You probably have to file a part-year resident return for the year you moved out of Virginia unless you moved on December 31. That part-year return will report income earned or paid while living in Virginia, and you will file a part-year return for the year you moved into Maryland that only reports income earned or paid while living in Maryland. You need to do the allocation yourself in Turbotax, the program can't guess for you. But after that, you would only file a Virginia non-resident return if you had Virginia income.)
On your federal return, you can claim a deduction for all state and local taxes (including property taxes) up to $10,000 if you itemize your deductions. Property taxes are deductible as long as the property is located in the US, it doesn't matter what state. If Maryland has a separate property tax deduction (unlikely) you would have to check the rules to see if the deduction applies to all property or only Maryland property.
