Praise be to Allah.
Mosques are the most honoured places on Earth; they are the houses of Allah and were built for His worship by praying, reading Qu’'aan, remembering Allah, seeking and teaching Islamic knowledge, and i‘tikaaf. Concerning this, the Prophet (blessings and peace of Allah be upon him) said: “… rather they [mosques] are for the remembrance of Allaah, may He be glorified and exalted, and prayer, and reading Qur’aan.” Narrated by Muslim (285).
Every Muslim has the right to enter the mosques and worship Allah there, so long as he does that in the prescribed fashion. Based on that, whoever wants to observe i‘tikaaf in some mosque has the right to do that, and no one has the right to prevent him from doing so. The one who prevents him may be included in the meaning of this verse (interpretation of the meaning):
“And who are more unjust than those who forbid that Allaah’s Name be glorified and mentioned much (i.e. prayers and invocations) in Allaah’s mosques and strive for their ruin? It was not fitting that such should themselves enter them (Allaah’s mosques) except in fear. For them there is disgrace in this world, and they will have a great torment in the Hereafter”
i.e., no one is more unjust than those who do this.
This prohibition is reinforced if it is done because these people who are observing i‘tikaaf are keen to follow the Sunnah and warn people against innovation and tell them not to cause mischief on earth, so restrictions are put on them to prevent them from observing i‘tikaaf in the houses of Allah.
Whoever strives -- in this case -- to prevent people from observing i‘tikaaf and staying in the mosques to pray, read the Qur’aan and remember Allah, is sinning and comes under the warning mentioned in this verse. The one who helps him to do that is like him.
As for the employees of the mosques, if they can take a stance for the sake of Allah against preventing people from observing i‘tikaaf in the mosques unjustly, this is what they should do, and they cannot afford to do anything else. If they cannot do that, then they should be gentle with the one who wants to observe i‘tikaaf and ask him to change his mind, because it is not wise for a person to observe i‘tikaaf in the mosque and thus cause harm to his Muslim brothers.
It may be better for him and for the Muslims to observe i‘tikaaf in the mosques where i‘tikaaf is permitted, so that a large number of Muslims will come together, and perhaps that will be a means of teaching them some Islamic rulings that they do not know, or one of them may receive sincere advice (naseehah) that will benefit him.
Prohibiting i‘tikaaf in some mosques may be acceptable if the mosque is small and does not have enough room for the worshippers, in which case observing i‘tikaaf there would bother the worshippers and make the place even more overcrowded.
And Allah knows best.