If a Mushaf has been given as a waqf to a specific mosque, it is not permissible to take it out of the mosque to read it at home or replace it with another one.
The scholars of the Standing Committee (16/19) were asked: Is it permissible to take the Mushaf out of the Haram to read it at home?
If Mushafs and books have been given as a waqf for people to benefit from in a specific place, it is not permissible to take them elsewhere, whether a Haram or somewhere else, unless the place where they were put is no longer in operation, in which case they should be moved to somewhere where they may be as beneficial or more so. End quote.
Standing Committee for Academic Research and Issuing Fatwas.
Shaykh ‘Abd al-‘Azeez ibn ‘Abdullah ibn Baaz, Shaykh ‘Abdullah ibn Ghadyaan, Shaykh ‘Abdullah ibn Qa‘ood
Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) was asked: Is it permissible to take a Mushaf from the mosque then return it?
It is not permissible to take a Mushaf from the mosque then return it, because the Mushafs that are found in mosques are awqaaf donated for the public benefit, for everyone who enters the mosque to benefit from. If someone takes them, then this implies that he is keeping them for himself and depriving others of it, which is haraam and is not permissible even if he replaces it with another Mushaf. The Mushafs in the mosques should be left as they are, and anyone who wants to read them may do so, in the mosque itself.
End quote from Fataawa ash-Shaykh Ibn ‘Uthaymeen, Noor ‘ala ad-Darb.
If there are many Mushafs in the mosque and they are not being made use of, there is nothing wrong with transferring the extras to another mosque that needs Mushafs.
Shaykh Ibn Baaz (may Allah have mercy on him) said:
If this small mosque does not need some of the Mushafs that are in it, there is nothing wrong with transferring those that are not needed in this mosque to another mosque that needs them, because the aim is that the worshippers benefit from these Mushafs. But to be on the safe side, it is better to ask the imam for permission to do that, because he knows best what the mosque needs.
End quote from Majmoo‘ al-Fataawa, 20/15.
For more information please see the answer to question no. 49886
To sum up: it is not permissible to take the Mushaf from the mosque or to replace it with another.
If the Mushafs in the mosque are more than is needed, the extra copies may be transferred to another mosque.
And Allah knows best.
Reproduced from Islam QA