Does active duty military pay VA state taxes for rental income earned in VA if state of residence is AK?

Yes, you will need to file a non-resident return for VA for the rental income.  See more information below:

Nonresident Military Personnel

Members of the armed forces who are not Virginia residents are not subject to Virginia income tax on their military pay, or on interest received from accounts held in Virginia banks. However, if you have other income from Virginia sources, such as a part-time job, or a business or trade conducted in Virginia, you are subject to tax on this income. You should report the income on a nonresident return, Form 763. Your residency status as a member of the military does not extend to your spouse or dependents. Non-military spouses and dependents living in Virginia are subject to the same residency and filing requirements as all other individuals.  https://tax.virginia.gov/military-tax-tips