What should I do if I knew of an unlawful relationship between a Muslim woman and a Christian man?

All perfect praise be to Allaah, The Lord of the worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu  ‘alayhi wa sallam, is His slave and messenger.

You should advise this woman either by sending her a message or through one of your Mahram women, if that is possible. If the woman does not accept the advice, you can inform her guardian in order to prevent this sin. The Prophet, sallallaahu ‘alayhi wa sallam, ordered the one who sees an evil to change it physically with his hand if he was able to do so. If he could not, he should change it by his tongue (by advising and/or condemning), and finally by (detesting it with) his heart, and he, sallallaahu ‘alayhi wa sallam, informed that this is the weakest level of faith. [Muslim]

In your case, you should change this evildoing with your tongue as you can not change it with your hand because you are not authorized to do so. However, if you were unable to change the evildoing with your tongue, you can detest it with your heart and this is the weakest degree of faith that is obligatory on all Muslims. It was narrated by ‘Abdullaah ibn Mas‘ood that the Prophet, sallallaahu ‘alayhi wa sallam, told us that the person who does not detest evil with his heart does not have faith. [Muslim] 

Allaah Knows best.