Is it permissible to offer expiation for fasting on behalf of an elderly man who does not comprehend fasting nor understand its meaning due to illness?

If a person is unable to fast due to old age or an illness from which recovery is not expected, he may break his fast, and he must pay a fidyah (expiatory feeding) of a poor person, based on the Almighty's saying: "{And upon those who are able to [fast, but with difficulty,] is a ransom of feeding a poor person.}" (Quran 2:184). However, if the illness is one from which recovery is expected, then he must make up the missed fasts when he regains the ability.

As for someone who loses his sanity: if it is due to fainting, he must make up the missed fasts. If it is due to insanity, then according to the majority of scholars, he is not required to make them up, based on the Prophet's (peace be upon him) saying: "The pen has been lifted from three: from the sleeper until he awakens, from the child until he reaches puberty, and from the insane person until he regains his sanity." No fidyah is obligatory upon one who has lost his sanity, whether due to fainting or insanity. Therefore, if the man mentioned in the question has a continuous loss of sanity, he is not required to make up any fasts nor to pay any expiation.