You can refuse the inheritance. In some cases you might have to file a disclaimer within 9 months. Regardless, the debt responsibility is not yours, unless you co-signed on the note. Often people inherit the homes, rent them while the foreclosure process is going on, and then abandon them with no liability. You should review your specific circumstances with an estate planning lawyer who is familiar with foreclosure defense.
Answer Applies to: Florida
I believe you can decline a specific bequest and if that is what the house is, then I believe you can pick and choose which one you wish to accept and which one you don't want to accept. Speak to an Estate lawyer about this matter as I'm certain there is case law available to check for a legal precedent. Good luck.
Answer Applies to: New York