The Lord’s Supper. If someone has not visibly, publicly identified with the body of Christ they should not be admitted to the family meal. Visiting Christians who are members of other faithful evangelical churches would be an exception to this, for the sake of charitably recognizing the body of Christ at large.
Any activity that publicly represents the church. This would include things like short-term mission trips, speaking at or hosting evangelistic events, and leading music. See the previous question and answer, “Must someone be a church member before being allowed to serve?”
Any teaching or leadership role. This would include things like leading a small group, teaching children, teaching adult Sunday school, or holding any church office.
Members meetings, if the church is congregational.
Small groups sponsored by the church (other than evangelistic small groups). There are at least three reasons to restrict small groups to members.
First, it helps nominal Christians who are given to casually grazing in whatever fields they please with no accountability understand that being a Christian means committing to the whole church, not just the niche groups which they favor.
Second, it helps Christians keep their eyes on the whole body, not just their favorite parts of it.
Third, it enables the groups to develop the unity and accountability that will only come when they are made up of church members. (None of this is to deny that Christians shouldn’t feel free to form small groups with Christians in other churches for the purposes of evangelistic outreach at their workplace, have thoughtful conversations about the social issues of the day, etc.)