Some mental hospitals are publicly funded through tax payer dollars and in that case you can usually go to those mental hospitals for treatment for around 24 to 48 hours without needing to pay out of pocket or have insurance to pay for the mental hospital stay.
Some other mental hospitals may be privately owned and funded and for those mental hospitals they would require you to either have medical insurance or have the money to pay the bill after you get out of the mental hospital.
But mental hospitals will accept you into the mental hospital either through involuntary commitment or voluntary commitment without needing to pay or provide medical insurance details.
Because most mental patients who need a mental hospital would be too distraught to sign any medical papers for insurance or payment.
They would bill you after the stay if they do bill you.
Some do bill mental patients while others do not and if the mental hospital is privately owned then they would bill you but not the publicly funded mental hospitals.