What is old earth creationism? What support does it have from the Bible? What mainstream denominations or groups, if any, support it?
Old Earth Creationism is the belief that the world was created but is contra Young Earth Creationism in terms of age. It does not specify an age, but instead assumes that current scientific dating is at least reasonably accurate.
If one does not literally interpret Genesis 1 in terms of days, we have an uncertain time since the creation of the world. If one does not interpret Paradise (which may have been created in 6 literal days) as the same world that we inhabit (consider other differences as well) then we do not have a certain date from the Bible other than the time at which Adam dwelled on Earth.
Most mainstream Protestant denominations at least support it, and it has support in both Roman Catholicism and Eastern Orthodoxy, though it is not considered dogmatic (you have Young Earth Creationists as well.) In general, groups which support Biblical Literalism will be more likely to consider Young Earth Creationism dogmatic or at least doctrinal, whereas other groups tend to leave it open to interpretation.
Among Church Fathers you have some who interpreted Genesis literally and assigned a Young Earth friendly age, whereas others spiritualized the days. The spiritualization of the days as a pattern in interpretation seems to originate with Origen, but probably predates him. Note that Origen is condemned as a heretic in Eastern Orthodoxy, while not all of his interpretive devices are, since non-condemned Fathers supported them.