If my late father told me not to get stuck with his debt, what can I do about his house if it has a mortgage on it?

You are not personally responsible for the debt--the house can be foreclosed on, if it is not paid, but you do not have to pay any of your personal money for it (unless you co-signed or guaranteed the debt, but you do not indicate that you did). Therefore:
1) If the house is worth more than the mortgage on it and you can pay it off, pay it off--that way you can keep the house, sell it for more than you paid on the mortgage, and make a profit.
2) If the house is worth same or less than the mortgage, let the bank take it.
You also, of couse, have to factor in other costs, like if you need to renovate, and the carrying costs until you can sell it. Example: house has $150k left on the mortgage, needs $35k of repairs to really be marketable, and will take 6 months to fix up and sell, during which you will pay $6,000 in taxes, utilities, etc. Therefore, to be able to sell the house means putting $191k into, and when you sell it, you have to pay 6% to a realtor. 
a) If the house is worth (after fixing up) solidly more than $250k, it's worth doing this (including paying the mortgage, assuming you can)--you should come out ahead.
b) If it's worth around $205k or so, you're break even; there's really no reason to do try to fix it up and sell it, unless you feel like you are in a strong housing market and may get an offer for more than it *should* be worth.
c) If it's worth $200k or less, let the bank take it.