Answer: Yes, though as you indicate, circumstances are important.
Some friendships are close and personal, others are more peripheral. Some friendships involve very personal sharing, whereas other friendships involve only a general acquaintance.
Clearly it is more possible to overlook many things with people with whom we are only acquainted, or with whom we simply have professional relationships. In these situations, our obligations to give and receive fraternal correction is less. But close friendships presume many shared values and similar respect for the truth. When such things are lacking in significant areas, close friendships are going to be strained.
Close friends also have greater obligations to instruct and admonish one another (cf. Jas 5:19; Gal 6:1). Hence, it is not the proper nature of a close relationship to simply overlook significant matters.
If I have a close friend and I know he is viewing pornography regularly or living with a woman outside of marriage, I have an obligation as a Christian to seek to correct him. If I have a close friend who is destroying his life with alcohol or drugs, I have obligations to admonish him and assist him to seek help.
All of these principles apply to someone with a homosexual orientation. If I have a close friend with this orientation and he or she is living celibately, this is fine, and I should seek to offer encouragement in this regard. If, however, they are straying into illicit sexual union and/or advocating a gay lifestyle, same-sex unions and so forth, I would have an obligation to instruct and admonish. It is difficult to see how a close relationship could continue if the individual were to utterly reject such correction about such a significant matter. The first concern for close friends ought to be each other’s salvation, not merely their feelings.
If a Christian were too weak to engage in this instruction, then it would seem that the close friendship is not really experienced as a friendship between equals, but a friendship wherein the other person has the upper hand. In this case, one might consider the admonition of Scripture that “bad company corrupts good morals” (1 Cor 15:33), and seek healthier friendships. For, once pressured to silence, many Christians give tacit approval, and the truth is no longer respected or proclaimed.