Depending on how you read the verse of wudhu (5:6 in the Qur'an) you can wash the feet or wipe on them. Sunnis wash the feet as in the 1st reading and wipe on socks in a state of wudhu as in the 2nd reading via the Sunni ahadith. I would like to see reports from the Shi'a where the Prophet (as) actually wiped on his bare feet please. Shi'a don't accept washing the feet at all I understand.
Shias use hadiths from Imam Baqir (علیه السلام) and Imam Sadiq (علیه اسلام) to support their position.
To quote a Shia source:
(باب) * (صفة الوضوء) * 1 - علي بن إبراهيم، عن محمد بن عيسى، عن يونس بن عبد الرحمن، عن أبان وجميل، عن زرارة قال: حكى لنا أبو جعفر (عليه السلام) وضوء رسول الله (صلى الله عليه وآله) فدعا بقدح فأخذ كفا من ماء فأسدله على وجهه ثم مسح وجهه من الجانبين جميعا ثم أعاد يده اليسرى في الاناء فأسدلها على يده اليمنى ثم مسح جوانبها ثم أعاد اليمنى في الاناء فصبها على اليسرى ثم صنع بها كما صنع باليمنى ثم مسح بما بقي في يده رأسه ورجليه ولم يعدهما في الاناء.
Zurara said: Abu Ja'far (Imam Baqir, علیه السلام) recounted to us the ablution of Messenger of Allah (ﷺ). So he asked for a bowl and took a palm of water and poured it on his face and then wiped all the two sides of his face. Then he returned his left hand into the bowl and poured it over his right arm and wiped over its sides. Then he returned his right hand into the bowl and poured it on his left arm and did to it what he did to his right arm. Then he wiped by what was left in hand over his head and feet and didn't return them into the bowl. (al-Kafi)
As you see there's no mention of washing the feet but wiping them by the water left on the hand. There are several more hadiths like this from al-Kafi.
This is debatable question. It all depends upon how you translate the word
famsahu means masah. Washing or just wiping(touching).
The verse in Quran is
(O you who believe! when you rise up to prayer, wash your faces and your hands as far as the elbows and wipe your heads and your feet to the (two) ankles. 5:6)
If you take the meaning of famsahu as washing, it simply implies that you should wash your head as well as wash your feet. Read it again, wash your head too if the word means washing.
But if you take the meaning of
famsahu as wipe that means you just have to wipe your head and wipe your feet[just touching].
What sunni brethern do is wipe the head but wash the feet and in quran, there is only one word used
famsahu. It either means wiping or washing. If it means washing, then you have to WASH both head as well as feet.
Regarding the comment below:
Yes, sunni brethern as well as shia brethern wash their faces. But for washing the word used is different. Here it is If God have asked to wash the feet but wipe the head then the
fagsilu word would have been used for washing feet as this has been used for washing the face and hands.