If my father dies and leaves his property to me and I do not sell it but decide to live in it, what kind of taxes will be due?

Unfortunately your father will die, no ifs or ands about it. He may leave the property to you if you are alive. Different states have different laws on inheritance taxation. The federal government also gets a share of estates over an amount which I can't remember, but it is more than a million dollars. The states that do tax inheritances have exemptions. Chances are there would be no taxes. You might want to have the house put in both your names by deed. When he is gone his interest vanishes leaving you the sole owner.

There is no federal tax on inheritances. For state tax, check at state web site.
Now your cost basis of the house is the fair market value of the house at the date of death; it is not the cost basis of your father.
For more information read: Your U.S. Tax Return: Tax on Inheritances

Putting your name jointly on the deed does NOT take the house out of your father's estate, but it WOULD make the probate process a LOT easier.

The federal exemption of estates is at $2 MILLION, and most states mirror that exemption level, so it is likely no estate or inheirtance taxes will be due. A call to the local state tax office will confirm or deny this assumption.