What is the ruling of Sharia on shunning relatives and friends who engage in illicit relationships, and is it obligatory to inform their families about these relationships?

Your advice to your friend and warning her against dealing with a foreign man is a good deed, and your friend's desire for marriage does not justify such behaviors. The mother sins by approving this relationship because she is assisting in sin, and it is her duty to restrain her daughter from it. If your friend repents from this relationship, then that is good; otherwise, avoiding her is better for you, according to the hadith: "Do not accompany anyone but a believer." Furthermore, shunning a disobedient person is permissible, and the benefit should be considered. If associating with the disobedient person causes harm to one's religion or worldly life, then shunning them is permitted. If the one you mentioned who permitted the relationship has erred, and his excuse of forgiveness is out of place, for Allah is severe in punishment. If they do not cease the relationship, then it is appropriate to inform the guardian of one of them. This falls under exposing corruption that must be reported to those in authority, and it is not considered part of concealing what is forbidden to reveal.