As a beneficiary, you inherit an asset at its probate value. If you then sell the asset and receive monies that reflect an increase in the value of the support, you will have to pay capital gains tax. The tax is paid on the difference between the value of the property when the previous owner died to when the property was sold.
If a beneficiary does not sell, they do not pay capital gains tax. In addition, beneficiaries can choose to live in or rent out the inherited property and not pay capital gains tax.