There is no federal inheritance tax, and the positive version of that statement is more useful than the negative one. Federal transfer taxes fall on the estate under chapter 11, on the giver under chapter 12, and on generation-skipping transfers under chapter 13. None of them is imposed on a recipient for receiving property. Going further, Internal Revenue Code section 102(a) provides that "gross income does not include the value of property acquired by gift, bequest, devise, or inheritance", so an inheritance is not federal income either. Income the property later produces is taxable in the ordinary way, but the receipt itself is not.
Whose state matters is the question these taxes are most often lost on. An inheritance tax generally follows the decedent's domicile, together with real property physically located in the taxing state. So an heir living hundreds of miles away in a state with no such tax can still owe it on a bequest from a relative who lived in a state that has one, and an heir living in a taxing state generally owes nothing on a bequest from a decedent who lived elsewhere. Moving after the death changes nothing.
The class structure is the design, not a detail. Where the tax exists it is typically built as a set of relationship classes, each with its own exemption and its own rate schedule. A surviving spouse is generally exempt. Beyond that, generalizing is unsafe: at least one state taxes direct descendants at a positive rate, and rates for siblings, nieces and nephews, and unrelated beneficiaries usually step up sharply. The practical consequence is that two people inheriting identical amounts from the same estate can owe very different tax.
Who actually files is not always who owes. The beneficiary is the taxpayer, but in practice the estate's executor or the financial institution frequently files the return and withholds the tax from the distribution. Deadlines are set by the state and do not track the federal estate tax's nine months. Some states discount the tax for paying within a few months of the death, and interest runs once a return becomes delinquent. The tax can also attach as a lien on real property, which is why a title company asks for a release before a sale closes.
It reaches beyond probate. Assets that pass outside a will, including jointly held property, payable-on-death and transfer-on-death accounts, and in some states certain life insurance and retirement accounts, are commonly within an inheritance tax's reach even though they never touch the probate estate. Avoiding probate and avoiding inheritance tax are two different projects, and the planning that achieves the first frequently does nothing for the second.