This matter has different opinions among the Maraaji' of Taqleed who are the highest in Islamic knowledge. Many of them e.g. Ayatullah Sistani and Ayatullah Sayyed Sa'eed al-Hakeem say that non Muslims are not allowed to enter Masjid according to obligatory precaution but they are allowed to inter all other religious places like Islamic centers and Husainiyya and Imam bargah etc.
'Other Scholars e.g. Ayatullah Kho'ee and Ayatullah Waheed Khurasani say that non Muslims are allowed to enter usual Masjids "except Masjid al-Haraam " if they don't carry with them any item which can make the Masjid Najis.