I inherited a house that has a mortgage and I am selling the house for what is owed and no profit will be made . Do I have to pay an inheritance tax

was it your principal residence- if so starting when?  were you a co-owner with the descendant? or was it lest vacant and held as investment property? or was the property rented to outsiders?

more info is needed. however, the mortgage is ignored. In determining gain or loss it's the selling price less your basis. 

Profit (Capital gain) is not the difference between selling price and mortgage balance.

The capital gain is the differnce between selling price and your cost basis.  Cost basis is usually what you bought the property for.  But, in the case of inherited property, your cost basis is the fair market value on the date of death.  This is known as the stepped up basis.   In most cases, there is no capital gain on property sold shortly after inheriting it.  So there is no income tax to be paid.   Considering the expenses of sale (e.g. real estate commission), there may even be a reportable capital loss (a tax deduction).

"Inheritance tax" (or estate tax) is something separate from the income tax on the sale of inherited property.  There is no inheritance tax, except of very wealthy estates ($12 million or more).  In addition,  six states also impose this tax: Iowa, Kentucky, Maryland, Nebraska, New Jersey and Pennsylvania.