My mother is deceased 1.5 years ago. Hospital has asked for $650 from her estate. The estate has been distributed, can I just ignore?

Hello - your question showed up to me as an attorney in Colorado.  I don't know if perhaps your mother passed away in Colorado, as I see that you are in Tennessee.  In Colorado a claim against an estate is barred (i.e., no longer valid) after one year following death (sooner if notice has been sent to the creditor).  If she passed in Tennessee, its laws would govern, but it may have a similar law.