As a non-citizen living in the USA, must I pay tax on any inheritance from my husband (also a non-citizen)?

No.

The rules on the payment of estate taxes can be complicated. Generally speaking, if both spouses resided in the United States, on the death of one of them there would be federal estate tax on that deceased person's world-wide assets. The resident alien would be entitled to the same exemptions (applicable exclusion amount) as a citizen would. As far as the marital deduction is concerned, the issues are more complicated. In general, if the surviving spouse is not a U.S. citizen, Section 2056(d) of the Internal Revenue Code would not permit a deduction for property passing to the surviving spouse unless the gift to that spouse was in the form of a Qualified Domestic Trust (QDOT). Neither would Section 2040(b) be applicable (Qualified Joint interest in property. So, the deceased spouse's estate would be subject to the estate tax without the off-setting of a marital deduction. These are general answers only. I don't know all of the particulars so my responses are qualified.

There is no inheritance tax [ on you] and normally no estate tax [on husband] unless the estate is over $5,000,000. Also, there is an unlimited exemption for a spouse however I can not remember if it is unlimited for an alien spouse. So my answer is probably no tax but see a lawyer to be certain.