My father in law inherited his parents home in Missouri. Will he have to pay taxes on it when sold and how is that figured? The house was bought years ago for $10k. And is now probably worth $120k. Does he need to get an appraisal within 30 days of death to set value on the house?

The estate would pay any taxes due prior to transfer. Once the property is in his name he pays taxes on the sale. The value of the property at the time his parents died (or whoever he inherited from) becomes his cost basis. If he sells for more then the cost basis he pays taxes on that amount.