Is my listening to my friend complaining about her husband's mother's treatment considered backbiting, and am I sinful for listening?

If the speaker's intention is to seek advice or complain, there is no harm, provided it is limited to the extent of need. However, if it is intended to defame or undermine the honor of her mother-in-law out of spite, then it is not permissible and constitutes backbiting (ghibah). It is not permissible for you to listen to her without refuting her; if you do not refute her, you will be sinful, because defending a Muslim's honor is obligatory, and a warning has been issued against neglecting it: "No man deserts a Muslim in a place where his sanctity is violated and his honor is assailed, but Allah will desert him in a place where he wishes for His help."