Is one who lies to his elderly mother who has lost her mental faculties considered to have erred? And is an expiation (kaffarah) obligatory upon them in place of her fasting and prayers?

There is no need to lie to a person afflicted with Alzheimer's. Rather, she should be told the truth, as there is no benefit or advantage in lying to her. As for prayer and fasting, there is no expiation or sin upon her, because she has become like a child who cannot distinguish, and therefore she is not held accountable.