Assalamu Alykum, This question is regarding Wudhū (ritual ablution). I have heard that if one does Wudhū once, completely and properly (let’s say at fajr time) then for the rest of the day, it is permissible for that individual to make masaḥ (wipe) over the top of one’s socks or shoes. If one wipes over their shoes, they must pray with their shoes on. If one wipes over their socks, one must pray with their socks on. I’ve also heard that opinion for women to do masaḥ over their ḥijāb. However, I came across this Ḥadīth: Narrated ‘Abdullāh bin ‘Amr: Once the Prophet (peace and blessings upon him) remained behind us in a journey. He joined us while we were performing ablution for the prayer which was over-due. We were just passing wet hands over our feet (and not washing them properly) so the Prophet (peace and blessings upon him) addressed us in a loud voice and said twice or thrice: “Save your heels from the fire.” (Saḥīḥ Bukhārī, Volume 1, Book 3 (Book of Knowledge), Ḥadīth 57) If you could please provide me with a better understanding of this issue, and whether it is permissible to do what I heard, and the message this Ḥadīth is conveying.
Jazāk Allāh Khayr /Thank you for your question.
Unfortunately the opinion that has been shared with you is an unsubstantiated one and hence any adherence to it must be avoided.
The issue of wiping socks or shoes in lieu of washing feet during ablution (Wudhū) is one that is often discussed and practiced without insight within certain circles in our community.
Washing the feet until the ankles while performing Wudhū is mandatory (fardh). Allāh the Exalted has instructed,
“O you who have believed, when you rise to [perform] prayer, wash your faces and your forearms to the elbows and wipe over your heads and wash your feet to the ankles.” [5:6]
It is for this reason Allāh’s Messenger (peace and blessings upon him) warned his companions against failing to wash their feet properly whilst performing Wudhū in the following incident;
‘Abdullāh ibn ‘Amr reported, “We returned with Allāh’s Messenger (peace and blessings upon him) from Makka [and were on our way] to Madīna. We came across some water on the way. Some people hurried at the time of ‘Asr; they performed Wudhū while they were rushing. [When] we reached them, their heels were apparently [dry]; water had not touched them. Allāh’s Messenger (peace and blessings upon him) proclaimed, “May the heels be ruined by the fire! Complete [your] Wudhū.” [Muslim: 570]
As for wiping one’s footgear in lieu of washing his feet during Wudhū, it is important to understand that a few types of footwear have been mentioned in various narrations upon which this action was apparently performed;
Khuff [a light leather sock]:
Shaykh Muḥammad Yūsuf Al Binūri has written in Ma` ārif Al Sunan, “Ibn Abī Shayba has reported in his Al Muṣannaf along with Ibn Al Mundhir in his Al Ashrāf and Ibn Daqīq al Eīd in Al Imam, “Hasan Al Baṣarī said, “Seventy companions of Allāh’s Messenger (peace and blessings upon him) have told me that they wipe [their] khuffs [during Wudhū in lieu of washing their feet].”
It is permissible for a non traveller (Muqīm) to wipe his khuffs with a wet hand when making Wudhū for 24 hours. The 24 hour period will begin from the time he renews his Wudhū for the first time after putting them on. A traveller (Musāfir) may wipe them up to 72 hours. Both are required to put them on after washing their feet for Wudhū.
Imam Muḥammad ibn Ḥasan Al Shaybānī (the student of Imam Abū Hanīfa) writes, “Wiping [the khuffs] is only permissible for every ritual impurity that necessitates Wudhū, not ghusl [a mandatory bath]. It is only permissible to wipe [them during Wudhū] when [a person] puts them on after completely purifying himself [by performing Wudhū]. If he was to wear his khuffs and then nullified [his Wudhū] before completing it, it is impermissible for him to wipe them [when renewing his Wudhū]…” [Al Mabsūt, Sarakhsi]
Mughīra ibn Shu’ba (may Allāh be pleased with him) related, “We were with Allāh’s Messenger (peace and blessings upon him) in a caravan and I had a container (of water) with me. He went to relieve himself and then returned. I confronted him with the container and poured [water out] for him. He washed his hands and his face and then attempted to take his forearms out [from the sleeves]. He had a woollen jubbah on that was made in Rome and had tight sleeves. They were too tight for him [to take his arms out] so he took them out from underneath.
I then bent down to take his khuffs off [so he may wash his feet] however he instructed, “Leave the khuffs [on] because I put them on my feet while they were pure.”
He then wiped them both.” [Abū Dāwūd: 151]
Khuzayma ibn Thābit reported, “The Prophet (peace and blessings upon him) said, “Wiping the khuffs is permissible for three days for a traveller (Musāfir) and one day and one night for a non traveller (Muqeem).” [Abū Dāwūd: 157]
Jawrab [a regular sock]
There are mixed views amongst the jurists (Fuqahā) regarding the permissibility of wiping regular socks whilst making Wudhū. It is important to note that the narrations pertaining to wiping regular socks are all weak (Dha’īf) and hence cannot be used to overturn or modify the directive of the Qur’ān.
Abū Dāwūd has recorded in his Sunan: Mughīra ibn Shu’ba reported,
“Allāh’s Messenger (peace and blessings upon him) performed Wudhū and wiped his jawrabs and na’ls.” 
He then writes, “`Abdur Raḥman ibn Mahdi would avoid relating this ḥadīth because the popular report from Mughīra is that the Prophet (peace and blessings upon him) wiped his khuffs. This account has also been reported by Abū Musa Al Ash’arī however its transmission chain is disconnected and weak.”
Surprisingly Tirmidhī has also recorded Mughīra ibn Shu’ba’s report on the Prophet (peace and blessings upon him) wiping his jawrab [ḥadīth no.99] in his Jāme’ with the exact same transmission chain as Abū Dāwūd and has labelled it as authentic. Regarding this Shaykh ‘Abdur Rahman Al Mubarakpūrī writes in Tuḥfat Al Aḥwadhī, the commentary of Jāme’ Tirmidhī, “Bayhaqī has recorded this ḥadīth of Mughīra’s and has said, “This is a Munkar ḥadīth. Sufyān al Thawri, `Abdur Rahmān ibn Mahdī, Aḥmad ibn Ḥanbal, Yaḥyā ibn Mo’īn, `Alī ibn Madīnī and Muslim ibn Ḥajjāj have all labelled this ḥadīth as weak (Dha’īf).”
He further states, “Nawawī said, “The scholars of ḥadīth unanimously agree upon its weakness, hence Tirmidhī’s assertion of it being authentic is unacceptable.”
In respect to the ruling on wiping regular socks, Imam Muḥammad writes, “As for wiping socks, it is permissible to wipe them provided they are thick and have leather soles.”
`Allāma Abū Bakr al Sarakhsi writes in the commentary of this statement, in Al Mabsūt,
“It is impermissible to wipe them [during Wudhū] if they are thin (such as sports socks and dress socks)…
According to Imam Abū Hanīfa, it is impermissible to wipe them if they are thick and do not have leather soles. This is because it is not possible to walk in them continuously [without tearing] while on a journey. Hence they will share the same ruling as thin socks.
According to Imam Abū Yūsuf (another distinguished student of Imam Abū Hanīfa) and Imam Muḥammad, it is permissible to wipe thick socks that do not have leather soles.
It is reported that Imam Abū Hanīfa wiped his socks during his illness and said to his visitors, “I have done what I have been preventing people from doing.”
They perceived this as a retraction of his previous view.”
Hence according to the Ḥanafi School of Jurisprudence it is permissible to wipe socks that are thick to the extent that one can walk a considerable distance in them without tearing.
Socks manufactured by Sealskinz and Gortex fit this criterion and hence can be wiped during Wudhū according to the conditions outlined for khuffs. Regular sports socks and dress socks do not fit the criterion and hence cannot be wiped.
Imam Muḥammad writes, “It is permissible to wipe jarmūqs [that are worn] on top of khuffs.” [Al Mabsūt, Sarakhsi]
Belāl reported, “The Prophet (peace and blessings upon him) would go out to relieve himself. [When he would finish] I would take water to him and he would perform Wudhū and wipe his turban and mūqs.” [Abū Dāwūd: 153]
In conclusion it is impermissible to wipe the shoes or socks that are commonly worn today as they do not fall under any one of the aforementioned categories.
As for women wiphjing their headscarf in place of wiping their head, Imam Muḥammad writes, “A woman cannot wipe her headscarf [while performing Wudhū].” [Al Mabsūt, Sarakhsi]
`Allāma Abū Bakr al Sarakhsi writes in the commentary of this statement, in Al Mabsūt,
“This is due to `Ā’isha’s ḥadīth in which it is reported that she put her hand under her headscarf and wiped her head [while performing Wudhū]. She then remarked, “This is what Allāh’s Messenger (peace and blessings upon him) instructed me to do.”
If however a woman wipes her headscarf and the wetness [of her hand] penetrates to her head to the extent that a quarter of her head becomes wet, it will be sufficient for her.”
And Allāh Knows Best.