It's been the standard practice of all the historic/traditional Churches since the beginning. Traditionally when a person was baptized they would also be anointed with oil (called chrism) and have hands laid upon them, this is known as Chrismation. In the West Chrismation underwent a bit of change because of peculiar liturgical and ecclesiastical oddities. Basically, in the West, it became customary for only bishops to administer Chrismation, and that often meant that a lot of time might pass between Baptism and Chrismation. Years, in fact. As such, during the late medieval period Chrismation became Confirmation. And since only baptized and chrismated persons could receive Communion, this led to "First Communion" happening for most Christians in adolescence, whereas in the East the standard practice is baptize, chrismate, and administer Communion all at once.
Lutherans, and I imagine other liturgically and traditionally oriented Protestants, have a complex relationship with these things. We Lutherans retained Confirmation as a "churchly rite" while not acknowledging that it is a Sacrament. Because the way we define the Sacraments is fairly strict and narrow. And, historically, we also post-poned First Communion to those who received Confirmation or were of a certain age. However, many Lutherans have come out and wondered what our justification is for doing this other than it's just how things have been done in the West for a long time. As such you'll find lots of Lutheran congregations that welcome baptized children and infants to partake of the Lord's Supper or, alternatively, parents may request that their children receive a blessing (with oil) in lieu of the Lord's Supper (the same as anyone else who is abstaining from receiving is welcome to receive a blessing).
Laying on of hands, also, is still the common practice involving ordination and installation of a new pastor. Speaking again from a Lutheran background, last Sunday we welcomed our new pastor. The presiding pastor of the synod, along with our interim pastor and our retired pastor laid hands upon our new pastor as part of the installation service.
Anointing the sick with oil, traditionally known as holy unction, likewise has been the historic practice of all the traditional and historic churches. While Lutherans don't regard Holy Unction to be a Sacrament, Catholics and Orthodox do. But regardless, it remains a key part of the Church's ministry in addressing the spiritual, emotional, mental, and physical needs of the Faithful.
Even animals receive blessings with oil sometimes, a common (though not universal) practice in many churches has been the Blessing of the Animals service held on the Feast Day of St. Francis of Assisi, where congregants are welcome to bring their pets to be blessed, as we are reminded that God cares for all His creatures, for He cares even for the nameless sparrows in the trees.
So, yes. Lots of laying on of hands and anointing with oil going on.
-CryptoLutheran