The Logic Behind covering The Face

In the total body of human being the most beautiful and excellent part is the face.  All the powers of hearing, seeing, speaking and all other powers, which create turmoil in the emotions and may lead men to sin are concentrated in the face.  That's the reason the Shariah has ordered women to cover the face with a nose-piece and should not allow any non-Mahram men to have a glimpse of her face in particular.

The Hadith describe severe warnings for a man who sees any non-Mahram woman with lust and desire.  The Holy Prophet (Sallallahu alaihi wa sallam) said:

Translation of Hadith: Whoever looks at a non-Mahram woman with lust, on the Day of Judgment, molten lead would be poured in his eyes.  (Hidayah, Kitab Ul Karahiyya, Vol. 4, Pg. No. 458)

If there is any possibility of desire, then everyone agrees that it is prohibited to reveal the face before non-Mahram people.  If there is no possibility of desire, then as per the Hanafi school of Fiqh, there is an allowance to keep the face uncovered.  Hence, in light of the clear corruption in this age, the later Fuqaha (Islamic Law experts) have stressed that the face should be covered.  Hence, while wearing the Burqa, it is more advisable and better to cover the hands up till the wrists, the ankles and including the face, the whole body should be covered.

As given in Radd Ul Muhtaar, Vol. 5, Pg. No. 261.

For whom is it allowed to see face

It is permissible for the Qazi (judge) and the witness to see the face of the woman.  In the same way, it is permissible to see the face of the woman whom one desires to marry.  There is a Hadith in Jame' Tirmidhi:

Translation of Hadith: The Holy Prophet (Sallallahu alaihi wa sallam) said: See the face of the woman whom you want to marry as this is a means of maintaining love.  (Jame' Tirmidhi, Hadith No. 1007)

At the time of proposing marriage, it is allowed only to see the face and the hand to the extent of the palms.  Seeing the face it is to get to know the facial features and the beauty and seeing the hands is to judge the condition of the body parts.  If more detail is desired, any reliable lady might be sent to know the other details about the girl.  In the same way, if the girl wants to see the boy for the purpose of marriage, the Shariah allows it.  As given in Radd Ul Muhtaar, Kitab Ul Hazr.

It is permissible for the doctor and/or the nurse to see the spot of illness/ailment of the patient.  Those parts of the woman's body, which are prohibited to be seen, they remain prohibited by the Shariah even if they are separated.  This is also the ruling for women's hair and nails of her legs.