Which Christian churches have an entirely unpaid/lay clergy?

Orthodox clergy, especially Russian Orthodox, are frequently not paid. Other Orthodox jurisdictions (e.g. Orthodox Church in America) may pay a small stipend, but frequently it is not enough to live on.

The Church of Jesus Christ of Latter-Day Saints (Commonly called "Mormons")

As a member, I enjoy paying tithing to the Church that is used to fund the building of the Kingdom of God on Earth: "Tithing funds are always used for the Lord’s purposes—to build and maintain temples and meetinghouses, to sustain missionary work, to educate Church members, and to carry on the work of the Lord throughout the world." (www.lds.org)

0% goes to any local leaders, 0% goes to church leaders. It all goes to the building of the kingdom. Everyone in leadership positions serve as volunteers and have careers on the side. Isn't it wonderful? :)

Coming from a Plymouth Brethren background, I can speak to the experience of growing up in a church movement which eschewed ordaining ministers and employing them as fulltime servants within local churches. Wikipedia has a decent overview of the Plymouth Brethren, a movement which began in the late 1820s in Dublin, Ireland and quickly spread to Plymouth, England in 1831.

". . . the central figures [in Dublin] were Anthony Norris Groves, a dentist studying theology at Trinity College, Edward Cronin, studying medicine, John Nelson Darby, then a curate in County Wicklow and John Gifford Bellett, a lawyer, who brought them together. . . . The first meeting in England was held in December 1831[5] in Plymouth. It was organised primarily by George Wigram, Benjamin Wills Newton and John Nelson Darby.[6] The movement soon spread throughout the UK. By 1845, the assembly in Plymouth had over 1,000 people in fellowship.[7] They became known as "the brethren from Plymouth" and were soon simply called "Plymouth Brethren".

Darby, by the way, has his own translation of the Bible as he was quite adept in biblical languages.

The Wikipedia article goes on to say,

"Brethren assemblies are led by the local church elders within any fellowship and there is no office of "pastor" among Brethren churches, because the term "pastor" is not found anywhere in the original koine Greek language of the New Testament. So, there is no formal ordination process for those who preach, teach, or lead, within their meetings. Men who become elders, or those who become overseers within the fellowship, are ones who have been recognized by others within the individual assemblies and have been given the blessing of performing leadership tasks by the elders. However, many assemblies also support full-time evangelists and workers."

"Conversely, an elder should be able and ready to teach when his assembly sees the "Call of God" on his life to assume that office (1 Timothy 3:2). As stated in the New Testament, Brethren elders conduct many other duties that would be typically performed by the clergy in other Christian groups, including: counselling those who have decided to be baptized, performing baptisms, visiting the sick and giving general spiritual advice. Some Open assemblies, especially the larger assemblies in North America, have salaried staff, including some designated as "Teaching Elders" or "Teaching Pastors."[citation needed] Normally, sermons are given by both the elders and other members of the meeting, but only men who the elders recognize have the call of God on their lives. Visiting speakers, however, are usually paid to cover expenses such as the cost of travel. Full-time missionaries are often financially supported by assemblies known by them, particularly their home assemblies."

When I was growing up, the Plymouth Brethren (PBs for short) did support missionaries both home (in the U.S.) and abroad, but those who served in the U.S. and Canada were called "fulltime workers." This meant that they were supported financially by one or more "assemblies" (the PB name for local fellowships around the world). For a number of years, the assemblies also had a mission organization stateside called Christian Missions in Many Lands (CMML), which published a magazine devoted to the work of PB missionaries throughout the world.

Another ministry of the PBs was, and still is, Emmaus Bible College, located in Dubuque, Iowa, which, in cooperation with other local colleges, grants four-year degrees in a number of disciplines and majors, including, of course, biblical studies. I attended the school for two years back when it was located in Oak Park, Illinois.

The PBs had as well a financial institution called the "Stewards Foundation" which helped to finance the planting of new assemblies by providing seed money for building a permanent house of worship, commonly called (in years past) chapels. I myself was brought up in Congress Avenue Bible Chapel in Rochester, New York, where I became a lay preacher and teacher for a number of years. Since I moved to Pittsburgh many years ago, the chapel building in which I grew up was sold and the members built and moved into a new facility in suburban Rochester. They called it Cornerstone Bible Chapel. It is doing very well, thanks to committed and Spirit-filled members, elders, and deacons who go above and beyond the call of duty to help grow the church spiritually.

In conclusion, the PB assembly movement has been on the decline for a number of years now, for a number of reasons. We needn't consider those reasons at this time. A new trend has emerged in the last few decades, however, and that is the financial support of elders. The biblical basis for this is 1 Timothy 5:17 NIV:

"The elders who direct the affairs of the church well are worthy of double honor, especially those whose work is preaching and teaching."

My own personal feeling (and I could be quite wrong) is that the spiritual vigor of the general membership in many PB assemblies has declined over the past 50 years or so. Worldliness in the form of materialism and a "let George do it" mentality have sapped the strength of many assemblies. The level of commitment that is required for a volunteers-only church to run at least at near-peak level has not changed in the last half-century, but that level has not been reached of late, particularly in those churches which lack a critical mass of unpaid but committed volunteers and leaders.

Paid fulltime elders and even paid staff members have helped to bring new vigor to some assemblies, but if a pervasive worldliness takes over in any given assembly, particularly to the point where there are simply not enough volunteers to bear the burdens of leadership and followership, the typical distinctives long associated with this once vital movement are likely to fade into oblivion.

Quakers and Mormons have no paid staff in their local assemblies. In the national bodies, however, there is usually some renumeration or support, in order to retain staff. The truth is that very few would consider it wise to have a national organization without any accountability, which is what an all-volunteer lobby would be.

Additionally, many more Baptists than you might think have volunteer staff. Finally, there are a great number of unpaid missionaries, itinerant preachers, and supporters of various causes that may fit into this category. Usually the local pastor is supported by the congregation, but the national staff has many more volunteers.

Finally, if you look at what a Roman Catholic priest earns, it's pretty close to nothing.

All that said, most mainstream Protestant churches will pay their clergy, based on Paul's commentary on paying preachers in 1 Timothy 5:18. Those churches relying on unpaid staff tend to be less mainstream and more "feeling" driven, with the feeling being that preaching is a gift like any other.