Praise be to Allah.
It is forbidden for Muslims to allow any kaafir to enter al-Masjid
al-Haraam [in Makkah] and the sacred areas [al-Haram] around it, because Allaah says
(interpretation of the meaning):
“O you who believe! Verily, the Mushrikûn (polytheists) are Najasun (impure). So let them not come near Al-Masjid Al-Harâm (at Makkah) after this year” [al-Tawbah 9:28].
With regard to other mosques, some fuqaha’ said that it is permissible because there is nothing to indicate that it is not allowed; others said that it is not permissible, by analogy with al-Masjid al-Haraam.
The correct view is that it is permissible if it serves the interests of sharee’ah or meet a valid need, such as a person hearing something that may invite him to enter Islam, or because he needs to drink water in the mosque, or the like. This is because the Prophet (peace and blessings of Allaah be upon him) tied up (his prisoner) Thamaamah ibn Athaal al-Hanafi in the mosque before he became Muslim, and the delegations of Thaqeef and the Christians of Najraan stayed in the mosque before they became Muslim. There were many benefits offered by doing this: they could hear the speeches and sermons of the Prophet (peace and blessings of Allaah be upon him), see people praying and reciting Qur’aan, and other benefits that are gained by those who visit the mosque. (From Fatawa al-Lajnah al-Daa’imah, 6/276).If some kuffaar ask to enter the mosque so that they can see how Muslims pray, so long as they have nothing with them that could make the mosque dirty, and there are no women among them who are dressed in a provocative fashion, or any other reason not to let them in, then there is nothing wrong with allowing them to enter and sit behind the Muslims so they can see how they pray. If we fear that there is someone who may rebuke them (for not praying etc.) without realizing that they are not Muslim, we should point this out to them.
And Allaah knows best.