If I inherit a house from someone who owes more than the house is worth, am I responsible for that debt?

Do you mean do you inherit it subject to the mortgage on the proeprty?  Yes, you do.  But you are permitted to keep on paying the present mortgage with out the bank calling it in and with out having to refinance the property. So if you can hold on and wait until the ptoeprty turns around and you can sell then that is an option.  Or else speak with some one about the situation and what you wish to do about accepting the inheritance.  Good luck.