If someone asks for mechila is it a hiyuv to mochel him, or just a very strong etza? Also, is it permitted to forgive someone from punishment, but not be friendly with them in the future—it seems there are many cases where a person is willing to forgive from punishment, but beyond them to have much of a relationship.
If the person asking mechilah is sincere, then there is a “moral requirement” to forgive him (see Rema 606:1). Gilyonei Ha-Shas (Yoma 87a) discusses whether or not this is an actual obligation, or only “worthy conduct,” and the latter option is the more likely: Forgiveness is something that depends on the heart, and it is not always possible to forgive; however, where it is possible, and the person in question is truly and sincerely penitent, it is worthy practice to follow the ways of Hashem, and forgive the penitent.
It is possible to forgive a person even when one is not interested in entereing into “much of a relationship” with them.
See also Sefer Hateshuvah 2:10; Shulchan Aruch Harav, 606:4.