Ok, from a purely legal perspective, if the son is the only child, and there was no will, he would be the only heir
to his father's estate. Since this is the case, if he is also appointed administrator that means that he is in charge of any assets in father's estate. The house would be an asset in the estate, so the son would be legally in control of that asset. This legally entitles him to collect any rents from it but also to pay any debts of the estate which include the mortgage and any taxes, insurance, maintenance, etc.
So to answer your question directly, yes, he as administrator has a legal duty to take control of the house as an asset of the estate and then use the rent to pay any expenses that are associated with its ownership. One of the primary duties of the administrator is to gather all assets, pay all debts and then distribute any surplus to the heirs at law.
As part of settling the estate, his duty would be to either sell the house to pay off any mortgage or transfer the house to himself and then refinance to pay off the debt and place the mortgage in his name alone. But he can't settle the estate until he can certify that he has paid all the debts of the estate.