I must admit I don’t follow these things too closely, but it seems many do.
I’m just not sure how widespread it is. I always see Amish with hats, for example but I’m not sure how significant it is to them.
As far as I know, the only faith that dictates headwear is Orthodox Judaism, for men.
There is a passage in one of Paul’s letters exhorting women to keep their heads covered, which is often taken by conservative groups as if a commandment. Offhand I think only the Amish and the more conservative Mennonites practice this on a day-to-day basis, though many older women and those in the more conservative churches will feel it obligatory to wear a hat to church.
Women in Islam are expected to remain modest (hijab) and among at least two schools of Shari’a interpretation that means, wear headgear that covers all the head. I’m not clear on how much of turban use is religious and how much cultural, though based on non-Arab Islamic states my impression is that it’s principally cultural.
Sikh men are expected to wear turbans, though I don’t know the context for that expectation and how mandatory it is.
What religions dictate headgear?
First Church of Christ, Orthodontist.
When entering a Catholic church, women should be wearing a head covering. Men are supposed to remove theirs.
–former parochial school boy.
Sikh men are required to wear a turban, because they are not allowed to cut their hair.
The women in the Apostolic Christian Church (not to be confused with pentecostal-like apostolic churches) also wear headcoverings. Small ones on a daily basis, and larger ones while attending church.
The church is a part of the Anabaptist religious movement, and the headcovering is part of that. Many liberal Mennonite sects no longer use headcoverings, but the Amish and conservative Mennonite sects still do, as well as some smaller sects like Apostolic Christian and Hutterites (someone else mentioned them as well).
The Behalt museum is devoted to the Anabaptist culture and is pretty interesting. They also have a large bookstore with books on all these religious traditions, and a room with samples of Amish clothing and hats, some of which you can try on, provided you are respectful.
Sorry if that was more information than you wanted. I live where the buggies leave ruts in the roads, and I was excited that someone asked a question I knew something about. 
Sikh men (and women) are not allowed to cut or shave or trim their hair (on any part of the body). Men are not required to wear a turban. That is more of a cultural tradition. There are well-known depictions of Sikh warriors going into battle with their hair flowing free.