The Prophet (peace be upon him) himself prayed from 610 for 13 years in a place where idols were.
This is the Holy Sanctuary of Mecca, which housed 365 idols that time. It was not just a simple sanctuary, but a place of pilgrimage for idolatry. And the Prophet (peace be upon him) is the pattern:

Ye have indeed in the Messenger of Allah a beautiful pattern (of conduct) for any one whose hope is in Allah and the Final Day, and who engages much in the Praise of Allah. (Quran 33:21)

Where worship is prohibited?

Narrated Sa'id: and the narrator Musa said: The Prophet (peace be upon him) said: “The whole earth is a place of prayer except public baths and graveyards.” (Sunan Abu-Dawud 492.)

Let's put together.

The prayer of a Muslim is performed for Allah and not for idols. No matter he/she is in a place where icons, depictions, statues are, his/her prayer is not directed towards those, but to Allah. If one does differently, he/she is committing a sin. Therefore, a Muslim can pray in the temple of any religion if he/she adheres to the above condition.

Although the Hadith only mentions baths and cemeteries, however in general it is not possible to pray in places where decomposing, rotting materials are and the conditions do not meet hygienic requirements. The cemetery is a resting place of decomposing bodies. There is no way for prayer here. This should not be confused with the supplication performed for the dead at a funeral. Nor can one pray in a temple that is built on a burial site.

There are situations when prayer is not only tolerated but also suggested in other religions' shrines. If there is an occasion it is nice to establish common prayer with them. For example, standing up for a common cause or mourning jointly. Following the Islamist assassinations against Charlie Hebdo in Paris, and killings of priests, Muslims have prayed in Christian temples together with their Christian brethren in many cases, condemning the brutal acts that happened.