Do I have to pay tax on $34000 inheritance from deceased father bank account?

The federal estate tax applies to estates with assets valued in excess of $5,000,000. The estate tax is not likely to apply. The threshold for paying an inheritance tax to the State of Illinois is also very high and not likely to apply. As the heir and legatee you do not pay any tax on distributions from an estate.

No- it is not income and any estate tax would have been paid by the estate.