The reason why tayammum is done on just two parts of the body

Ibn al-Qayyim (may Allah have mercy on him) said: 

The reason why tayammum is only done on two parts of the body (i.e., the face and arms) is that this is quite appropriate, based on analogy and common sense. For placing dust on the head is something that is ordinarily disliked, rather it is customarily done at times of disaster and calamity, and the feet are usually in contact with the dust anyway. But putting dust in the face is an action of humility and veneration of Allah, submission to Him and humbling oneself before Allah, which is one of the most beloved of acts of worship to Him and one of the most beneficial to the person himself. Hence it is mustahabb (liked, encouraged) for the one who is prostrating to get dust on his face for the sake of Allah, and not to protect his face from getting dusty. One of the Sahaabah saw a man prostrating and putting something between his face and the dust, and he said to him: “Get your face dusty.” This meaning does not apply in the case of getting the feet dusty. 

Moreover, this is appropriate by analogy in another sense, which is that tayammum applies to the parts of the body which are washed during wudoo’, but not to the parts which are merely wiped. For the feet may be wiped when one is wearing socks, and the head may be wiped when one is wearing a turban. Since the parts which are usually washed are reduced to wiping (in the case of tayammum), the parts which are wiped are reduced to nothing, because if we were to wipe the latter with dust, that would not be a reduction at all, rather instead of wiping them with water we would just be wiping them with dust. Thus it becomes clear that what sharee’ah tells us is the most fair and the most perfect.  

The tayammum of the one who is in a state of junub (major impurity following sexual activity) is like the tayammum of the one who is in a state of minor impurity. If the one who is in a state of minor impurity is spared the obligation of wiping the head and feet with dust, then it is more appropriate that the one who is in a state of major impurity should be spared the obligation of rubbing his whole body with dust, because that would involve a great deal of hardship, and it goes against the idea of tayammum being a dispensation. It would also reduce the one who is the noblest of Allah’s creation to becoming like the animals who rub their bodies in the dust. That which was brought by sharee’ah cannot be superceded in wisdom and fairness, and to Allah be praise.