Whether covering a women's face is wajib has been debated for a long time. Since the side that assures that it is wajib has already been covered, I'll cover here the side that says it is not. I will be referring to the writings of Sheikh Al-Albani, who is considered by many as the most prominent Sunni scholar in modern times to have come to criticize covering the face as being obligatory.
When it comes to the ayah in the question (Al Nur 24:31), the part that is in dispute is this:
وليضربن بخمرهن على جيوبهن
The two crucial words are خمرهن (khumurihin) and جيوبهن (jiyubihin). According to Al-Albani, the first word is the plural of خمار (khimar) which he translates as head covering and the second word is the plural of جيب (jayb) which is the collar line (the area where a shirt/dress is cut off to expose the neck). So in this translation the ayah does not instruct Muslim women to cover their faces, but only their heads (as in their hair).
Then we have Al-Ahzab 33:59
O Prophet! Tell your wives and your daughters and the women of the
believers to draw their cloaks (veils) all over their bodies (i.e.
screen themselves completely except the eyes or one eye to see the
way). That will be better, that they should be known (as free
respectable women) so as not to be annoyed. And Allaah is Ever
Oft‑Forgiving, Most Merciful
The point of contention is the translation of the verb يدنين (yudneen), that many translate (as in the above translation) as "to cover the face (with a veil)". Al-Albani postulates that the verb actually means "to bring closer" and does not specify the face. And the word جلابيبهن (jalabeebihin) is the plural of جلباب (julbab), which means "covering" without any specifics as to what it covers.
A third ayah, Al Ahzab 33:53
O you who have believed, do not enter the houses of the Prophet
except when you are permitted for a meal, without awaiting its
readiness. But when you are invited, then enter; and when you have
eaten, disperse without seeking to remain for conversation. Indeed,
that [behavior] was troubling the Prophet, and he is shy of
[dismissing] you. But Allah is not shy of the truth. And when you ask
[his wives] for something, ask them from behind a partition. That is
purer for your hearts and their hearts. And it is not [conceivable or
lawful] for you to harm the Messenger of Allah or to marry his wives
after him, ever. Indeed, that would be in the sight of Allah an
The word حجاب (hijab) here means screen, and the ayah itself only concerns itself to conduct when entering a house (the prophet's house in particular) and not for day to day veiling.
Furthermore, Al-Albani cites many hadith, such as:
It was narrated that Jabir said: "I attended the prayer with the
Messenger of Allah () on the day of 'Eid. He started with the prayer
before the Khutbah, with no Adhan and no Iqamah. When he finished the
prayer, he stood leaning on Bilal, and he praised and glorified Allah
(SWT) and exhorted the people, reminding them and urging them to obey
Allah (SWT). Then he moved away and went to the women, and Bilal was
with him. He commanded them to fear Allah (SWT) and exhorted them and
reminded them. He praised and glorified Allah, then he urged them to
obey Allah, then he said: 'Give charity, for most of you are the fuel
of Hell.' A lowly woman with dark cheeks said: 'Why, O Messenger of
Allah?' He said: 'You complain a great deal and are ungrateful to your
husbands.' They started taking off their necklaces, earrings and
rings, throwing them into Bilal's garment, giving them in charity."
(Sahih) Sunan Al-Nasa'ai
It is apparent here that since the narrator was able to describe the woman's cheeks that her face was uncovered in public. Something that was not commented on negatively.
Yahya related to me from Malik from Yahya ibn Said from Amra bint Abd
ar-Rahman that A'isha, the wife of the Prophet, may Allah bless him
and grant him peace, said, "The Messenger of Allah, may Allah bless
him and grant him peace, used to pray subh and the women would leave
wrapped in their garments and they could not yet be recognised in the
Al-Albani deduces that since it was the darkness (and only the darkness) that lead to the women becoming unrecognizable, then it comes to reason that they are recognizable in the light, which means that their faces were known and seen.
One more hadith is this one, although it's authenticity is disputed:
Narrated Aisha, Ummul Mu'minin:
Asma, daughter of AbuBakr, entered upon the Apostle of Allah ()
wearing thin clothes. The Apostle of Allah () turned his attention
from her. He said: O Asma', when a woman reaches the age of
menstruation, it does not suit her that she displays her parts of body
except this and this, and he pointed to her face and hands. Abu Dawud
said: This is a mursal tradition (i.e. the narrator who transmitted it
from 'Aishah is missing) Khalid b. Duraik did not see 'Aishah.
Sunan Abu Dawud
This hadith has direct instruction on what not to cover. But, this hadith is weak according to Abu Dawud (see his commentary at the end of the quote), and most scholars. However, both Al-Albani and Al-Bayhaqi have disputed this classification.
The following pages (Arabic) have more on Al Albani's writings in the subject: