If your father or your family is personally paying for his nursing home care, he can do whatever he wants with his house – sell it, give it away, keep
it and rent it out. However, if he keeps it or sells it, the value of the house will be counted against his potential eligibility for Medicaid coverage of his nursing home costs. He'd have to spend nearly all the proceeds from the sale of the house before he'd be eligible for Medicaid. If he gives away the house to you or anyone else, its value will still count against his eligibility for the next 60 months following the gift.
If your father is still competent to make financial decisions, he should execute a power of attorney for finances so that you or another family member can act on his behalf in transactions relating to the house. The document should specifically be designated a "durable" power of attorney so that it remains valid if and when your father becomes mentally incapacitated.
If your father is already mentally incapacitated and he has not previously executed a durable power of attorney for finances, your family may have to go to court to have a judge appoint you or another family member to be your father's conservator or adult guardian. This person can be empowered by the court to make financial decisions relating to the house, as well as your father's other assets.