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

Strikes me as a variation of Slippery Slope (also known as absurd extrapolation, thin edge of the wedge, camel's nose, domino fallacy)

Description: When a relatively insignificant first event is suggested to lead to a more significant event, which in turn leads to a more significant event, and so on, until some ultimate, significant event is reached, where the connection of each event is not only unwarranted but with each step it becomes more and more improbable. Many events are usually present in this fallacy, but only two are actually required -- usually connected by “the next thing you know...”