As for the face, it may not be wiped—except in the unusual case where a splint is in place, that is, a dressing laid over a wound or the like. If someone were to cover his face with a cloth to shield it from the burning sun or for some similar reason, he is not allowed to wipe over that covering; he must remove it and wash his face. Only when there is a genuine necessity may he substitute wiping over what covers his face in lieu of washing.
The same applies to the hands: they are not wiped; they must be washed—unless an unavoidable situation arises, such as a skin condition that would be harmed by water, in which case the hands are wrapped, or gloves are worn to keep water from reaching them. In that circumstance, wiping over the wrapping or gloves takes the legal status of wiping over a splint, and is permitted because of the necessity.
The head, by contrast, is to be wiped, and its purification is lighter than that of the other limbs. Hence, if a woman’s head is coated with a thick layer of henna, or a pilgrim has his hair matted with gum, as the Prophet ﷺ did while in the state of iḥrām, it is sufficient to wipe over that coating; there is no need to remove it.
As for the feet, they may be washed or wiped, and for this reason the noble Qur’an presents both possibilities in its recitation of the phrase wa-arjulakum / wa-arjulikum. One reading has the accusative vowel (arjulakum), and another has the genitive (arjulikum). The genitive reading links the feet to His words, “and wipe over your heads,” meaning: wipe over your feet. The accusative reading links the feet to His words, “wash your faces,” meaning: wash your feet.