It depends. Since your spouse is a VA resident and you are not (but the income is VA income), Virginia requires you to file separate tax returns. How you do so, however, is up to you to determine. She could claim all of the rental using the Married Filing Separate status on a resident return. This would keep the rental income off of an Illinois return because you are not required to file a joint return in Illinois as she is not an Illinois resident. This is probably the simplest treatment.
If you file any portion of the rental as a Virginia nonresident, Illinois will tax that income also. Although your military income is not taxable in Illinois, the rental income would be. Illinois would give you a credit for any tax you pay to Virginia on the income, but it would be two separate tax returns.
Based on this, my recommendation would be to file the rental to Virginia under your wife's income using the status Married Filing Separate, if for no other reason than it is less complicated. Having said this, the final decision is up to you.
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