If an active duty military person sells a house outside of their state of residence, do they pay taxes in their official state of residence or where the house was?

Normally you would pay taxes to both states.  A capital gain from the sale of real estate located in State X is taxable by State X even if the taxpayer is not a resident of State X.   (In fact many states require that state income tax be withheld at closing when property is sold by a non-resident.) 

That capital gain would also be taxable by the taxpayer's State of Residence, which can tax ALL the taxpayer's income.  

In such a circumstance, the taxpayer could take a credit on his resident state return for the taxes paid to the non-resident state, so there would not be double taxation.