A person can only have one domicile at a time. Domicile is a person's true permanent home, there is no single factor that determines, but some key factors include the location of a house, car registration, voter registration, professional relationships like doctor and attorney, and significant social relationships. Intent is also a key factor. Most importantly for you, establishing a new domicile requires actively taking steps to abandon a prior domicile.
On one hand, it is possible to live away from your domicile for a long time without actually changing it. And being unwilling to sell the existing home could be an important factor in showing that your father is still domiciled in his original state. On the other hand, if you and your father have both acknowledged that he is now permanently living with you, and he has changed his doctors and other legal relationships, that may be enough to show he has changed his domicile even though he refuses to sell the house. What is his intention? Does he intend to return and live independently some day (even if that is medically unrealistic to you)?
Only you and he can analyze all the factors.
If he is domiciled in state A but living full time in state B, he still files a resident tax return in state A that reports and pays income tax on all his world-wide income. He would only file a non-resident return for state B to report "state B-sourced" income. State sourced income includes things like wages earned in state B and real estate income in state B, but does not include a pension or investment income that happens to be paid to him while living in state B.
However, some states have a "statutory resident" rule. That means that even though the person is domiciled in state A, they will be treated as a resident of state B, and will be required to file a state B resident income tax return that pays tax on all their world-wide income, in addition to a state A resident income tax return. In this case, state A may be unwilling to give the usual credit for taxes paid to a non-resident state.
To know the full implications, we need to know which two states you are referring to.
It may be that if your father intends to remain domiciled in state A but lives all year in state B, he will owe double state income taxes. The only way to fix that would be to abandon the domicile in state A and make state B his new domicile. This would not necessarily require selling the house, but at least changing the mail and taking other steps such as moving key items of personal property from his house to yours, to show intent to permanently change domiciles. I realize this may be difficult. You will need to evaluate his situation and make your decisions as best you can.
