Can my sister sell my mother's house while her brother still lives there?

He would need to be a legally recognized tenant in order to have standing otherwise he can be evicted. "Legally recognized" where I live means he has a rental agreement that was filed / recorded at the courthouse, & so as long as he does his part (like pay rent), he can stay there under whatever terms done even if property gets sold. If you refuse rental payment, he still can stay assuming he sends it to you via certified mail or other verifiable delivery. If nevertheless you want him out, your going to have to pay him enough to make it worth his while to move out.

But if not then you do paperwork with sheriffs Dept to have him evicted.
The buyers may include for a clause in the act of sale that the seller has tenants removed in order for sale to go through.
Not pretty either way.

What does mother want? Is she still capable of making such decisions? Has she been declared legally incompetent? Has a doctor said that she is?

Who is living in her house? Her brother or her son? Was he living with her before she went into the NH? How long?

Is Mother on Medicaid? Is her money running out?

Sorry for so many questions, but the answers will help ensure more meaningful answers.