If a father dies and leaves a child money as the beneficiary on an account, is the beneficiary responsible for medical bills once the person dies?

You would be very well advised to have everything looked at to avoid an error, but generally when you are named as a beneficiary you receive the funds by contract from the source, not through the estate. So, while your father's estate remains responsible for his bills before any distribution can be made to heirs or legatees, the funds you describe may not belong to the estate. For example, if you are named beneficiary of a life insurance policy, the funds are payable to you from the life insurance contract. Your use of the word "account" suggests, however, that it may not be as clear cut as this. Take the paperwork to any attorney so that you do not land yourself in a costly mess down the road.