Question: People are saying that if you're going to let immigrants in you must be willing to open your house to them.

You can use an analogy to show how nonsensical this is. You can ask, "If we allow people to give up children for adoption in the United States, then must be willing to adopt 3 or four children? If not, why the difference?" I don't think there is a fallacy per se, but I do think an unreasonable claim is being made.