Does "spare their feelings" mean "share their feelings"?

No.

To "spare" someone's feelings means to "protect" someone's feelings. It means to do something with the intent of preventing someone's feelings being hurt.

Often, sparing someone's feelings involves lying or deceiving them in some way. In this case, it is laughing at someone's humor... even though it isn't funny so that they won't be hurt by the fact that they aren't really funny.

Cool.
Thanks.
BTW, is "install" used properly in the following sentence?

The mistress of the house reluctantly installed the two refugees in a spare room.

Yes.

I don't think you install refugees.

When install is used to refer to human beings, it generally involves putting someone into an office (i.e. a job for them to do). You can install a priest into a parish... but you can't install a refugee.

In British English usage it is fine to write of "installing" a person somewhere as in the original question.

That's interesting.

Some of the meanings of "install" I found in dictionaries are

"to establish in an office, position, or place", e.g. to install oneself in new quarters.

"To settle (a person, esp oneself) in a position or state" e.g. she installed herself in an armchair.

"After six months in a psychiatric ward, I decided she would stay with me. I had been living with a man named Steven for a couple of years in the late `60s. We would install mother in the extra bedroom, and I would work on her diet, and she would continue seeing the psychiatrist, and we would be together. I remember thinking, ``Now I`ll see if I have a mother.``

The Long Way Home: A Daughter Charts Her Mother`s Mysterious
Journey Out Of Addiction. November 13, 1988 By Robin Herndobler (Chicago Tribune)