Praise be to Allah.
The muhrim (pilgrim in ihram) is forbidden to deliberately remove hair from any part of his body, whether from his head, his beard or elsewhere, because Allah, may He be exalted, says (interpretation of the meaning):
“And do not shave your heads until the sacrificial animal has reached its place of slaughter”
It is permissible for the muhrim to rub his hair and beard, and to run his fingers through them gently, if he is certain that he will not make his hair fall out by doing that.
The evidence for that is the report narrated by al-Bukhaari (1840) and Muslim (1205) from ‘Abdullah ibn Haneen, according to which ‘Abdullah ibn al-‘Abbaas and al-Miswar ibn Makhramah that they had a difference of opinion in al-Abwa’. ‘Abd-Allah ibn ‘Abbaas said: The pilgrim in ihraam may wash his head, and al-Miswar said: The pilgrim in ihraam may not wash his head. Ibn ‘Abbaas sent me (the narrator) to Abu Ayyoob al-Ansaari (to ask him about that), and I found him washing himself between the two poles of a well, screened with a cloth. I greeted him with salaam and he said: Who is this? I said: I am ‘Abd-Allah ibn Hunayn. ‘Abd-Allah ibn ‘Abbaas has sent me to you to ask you how the Messenger of Allah (blessings and peace of Allah be upon him) used to wash his head when he was in ihraam. Abu Ayyoob (may Allah be pleased with him) put his hand on the cloth and lowered it until his head became visible, then he said to the person who was pouring water for him: Pour some water. He poured it onto his head then he rubbed his head with his hands, moving them forwards and backwards. Then he said: This is what I saw him (blessings and peace of Allah be upon him) doing.
Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) said in his commentary on the hadith of al-Miswar and Ibn ‘Abbaas:
This indicates that it is permissible for the pilgrim in ihram to wash his head and run his fingers through his hair.
End quote from Saheeh al-Bukhaari (p. 20).
If the pilgrim in ihram fears that some hair may fall from his beard, then he should run his fingers through it gently, or rub it with a cupped handful of water so that the water reaches all of it.
Shaykh Ibn ‘Uthaymeen said:
Causing the water to reach the inside of the beard may be done in two ways:
Taking a cupped handful of water and placing it under the beard rubbing the beard in it until the water reaches all of it.
Taking a handful of water and running his fingers through it like a comb.
See: ash-Sharh al-Mumti‘ (1/173).
Ibn Hajar said in the context of discussing what we learn from the hadith of al-Miswar and Ibn ‘Abbaas quoted above:
It is permissible for the pilgrim in ihram to wash his hair and run his fingers through it, and rub it with his hand, if he is certain that no hairs will fall out.
He quoted as evidence for that the fact that running the fingers through the hair of the beard when doing wudoo’ remains mustahabb (encouraged), in contrast to those who say that it is makrooh (disliked), such as al-Mutawalli among the Shaafa‘is, for fear of causing hairs to fall out, because it says in the hadith: “then he rubbed his head with his hands.”
There is no differentiation between the hair of the head and that of the beard, except that it may be noted that the hair of the head is stronger and less likely to fall out.
Through examination of the matter, we may conclude that what is more appropriate will vary from one person to another. This was stated by as-Subki al-Kabeer. And Allah knows best.
End quote from Fath al-Baari (4/57).
Al-Khateeb ash-Sharbeeni said:
The apparent meaning of the author’s words with regard to the Sunnah of running the fingers through the beard is that there is no differentiation between the pilgrim in ihram and anyone else, and this is the correct view, as stated by az-Zarkashi in his Khaadim, in contrast to the view of Ibn al-Maqri in his Rawd, in which he followed al-Mutawalli.
But the pilgrim in ihram should run his fingers through his beard gently, lest he cause any hair to fall from it, as they said concerning running the fingers through the hair of the deceased [when washing him prior to burial].
End quote from al-Iqnaa‘ fi Hill Alfaaz Abi Shujaa‘ (1/50).
To sum up: it is prescribed for the pilgrim in ihram to run his fingers through his hair and beard when doing wudoo’, and to do so gently.
If he fears that some hairs may fall out, then he may wash his hair and beard with a cupped handful of water, without combing through it with his fingers.
If any hairs fall out, without him doing anything that causes them to fall out, there is no blame on him, because he did not deliberately remove some of his hair.
Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) said:
Whatever hairs fall from the head and beard of the pilgrim when he does wudoo’ or ghusl, without him intending to do that, is pardoned, because it is dead hair.
This ruling also applies to the one who wants to offer a sacrifice (udhiyah) after the first ten days of Dhu’l-Hijjah begin.
Rather what is forbidden is to deliberately cut any of that hair when in ihram or after the first ten days of Dhu’l-Hijjah begin, for the one who intends to offer a sacrifice (udhiyah).
End quote from Fataawa Islamiyyah (2/713).
For more details on the ruling on running the fingers through the beard for one who is not in ihram, please see the answers to questions no. 85031 and 110261.
And Allah knows best.