Are non-Muslims allowed inside mosques?

Yes.

The Holy Quran does not prohibit anyone from visiting a mosque (provided they are not there to cause trouble) as it is a House of God that can and should be used for the worship of God by all. Only idolatry is prohibited in a mosque but even then all are welcome to visit a mosque. This declaration against idolatry in a mosque was to protect the Ka’aba (and therefore any mosque) and to ensure that it would remain a sanctuary for those who believe in the Oneness of God.

According to Islam non-Muslims are even allowed in the Sacred Mosque in Makkah and Madinah.

It is recorded that the Christians of Najran came to see the Holy Prophet Muhammad^(sa). He arranged the meeting in his mosque at Madinah, during the meeting the Christians asked leave from the mosque for worship. The Prophet^(sa) said that the mosque in which they were was a house of God and they were welcome to offer their prayers there. So they did offer their prayers in the mosque of the Holy Prophet^(sa). (Ibn Hisham, I, 575-577)