His "partner" has dumped his funeral plans in my lap. I can, and will, take care of that , but don't want to open myself up to any medical bills or other debts. He has adult children but don't know where they are. I am in Missouri.
Estate Planning Attorney
In short, no. And if you somehow receive correspondence or phone calls, you can ignore them.
Your brother's estate is responsible for his debts. But you are not responsible to find his Will and file probate papers in the local county where he died if you are not the named executor (if he had a Will), nor do you have a reponsibility to file papers to become appointed administrator of his estate (if he did not have a Will). Still, someone should take charge -- such as one of his children, since they would stand to gain under his state's intestacy laws -- especially if there are assets left in his name which would exceed the debts. If his estate would be zero or negative, you might at least want to keep a few death certificates around in the event collectors come knocking on your door, to prove to them that he's gone.
However, my answer is only based on the limited information you provided. There may be certain circumstances that make it very necessary and advantageous for you to take charge -- even if only to see that your brother's unfinished work gets satisfactorily completed. You might want to find an attorney for a consultation, bringing in any paperwork you know of. But most likely, you can leave all of this to his children to take care of.