Is it permissible for a Muslim to get involved in the construction of the house of worship of other religions?
Scholars have two differing opinions on this matter. Some of the scholars in madhhab Hanafi ruled it permissible for Muslims to be involved in the construction as well as working in a church. Among the scholars who presented this opinion are Ibn ‘Abidin  and Ibn Nujaim . The majority of scholars in madhhab Maliki and Syafie prohibit it. This could be further analyzed from the opinion of Imam Malik himself , Imam Muhammad al-Hattab , Imam al-Syafie , Syeikh Qalyubi . Whereas, Ibn Hajar al-Haitami stated that if a Muslim give wakaf on sinful matters such as building a church, then the wakaf is invalid.  Imam Ibn al-Qayyim prohibited renting a room or building that is going to be made into a church or a house of worship for the Zoroastrians. 
Syeikh Dr Yusuf al-Qaradhawi issued a fatwa stating that it is permissible for a Muslim to be involved in the building of a church. He was asked regarding the ruling for a Muslim to be involved in the bidding/accepting a tender for the construction of the house of worship for other religions and the ruling of benefitting or getting a profit out of the project. He drew an opinion from the opinion of some of the scholars of madhhab Hanafi who permitted the involvement of a Muslim in building a church or the house of worship of other religions, although he also reminded that more scholars ruled it prohibited compared to those who permit it. 
I’m inclined towards the opinion which prohibits the involvement of Muslims in the building of a church or other religions’ house of worship, as held by the majority of scholars. Allah SWT states:
وَتَعَاوَنُوا عَلَى الْبِرِّ وَالتَّقْوَىٰ ۖ وَلَا تَعَاوَنُوا عَلَى الْإِثْمِ وَالْعُدْوَانِ
“Cooperate with one another in goodness and righteousness, and do not cooperate in sin and transgression.” 
However, Islam is a wide and flexible religion as well as a religion that celebrates the needs of people where it also provides exemptions in certain situations. For example, if a person won a tender to build or renovate another religion’s house of worship, it is obligatory for him to reject the tender. However, if by doing so he’ll be liable to pay a fine or compensation which will burden him, then he is allowed to taqlid to madhhab Hanafi and fatwa from Syeikh Yusuf al-Qaradhawi who permitted it based on maslahah.
If a person works in a company that is involved in the building of another religion’s house of worship, he should discuss with his employer to exclude him from being involved in the work. However, if this isn’t possible, and he feared that he may be laid off if he disobeyed his employer’s orders, furthermore, if he is in a desperate situation where people are trying to cutting on costs and is laying off workers, then it is also permissible for him to taqlid to the madhhab which permits him to do so based on the exigent situation. Wallahua’lam.