Register of Copyright Mary Beth Peters, with whom I worked as a fellow Policy Planning Advisor from 1988 to 1992, has made a number of very important improvements in the operation of the Copyright Office. For example, she abandoned the Office's former refusal to accept for registration a work that would be registrable but for the fact that a prior design patent had been obtained. And she instituted short form registration to simplify matters. Another improvement is the institution of a formal internal appellate procedure for refusals to register claims, explained most recently in a December 28, 2004 Final Rule in the Federal Register and placed (although not yet, apparently from the Office's website) in 37 C.F.R. Section 202.5. The Office doesn't call these appeals, but rather "requests for reconsideration," but because it is shorter I will call them "appeals." The history of such appeals is spotty.
In the days before there was a Copyright Office, occasional appeals from a refusal to register by the Librarian of Congress were taken to Congress's Joint Committee on the Library, which held oral argument. In the mid 1930s, within the Copyright Office, there was a "Revisory Board" consisting of three members who issued written opinions on a wide variety of topics. My favorite is one where two members of the panel thought the work of graphic art depicted a typical Pennsylvania Dutch holiday scene but the third deemed it obscene. Double entendres were also an issue of deep concern to the Revisory Board, including one involving a song entitled "Santa has Cotton Balls and a Candy Cane."
But when the Revisory Board disbanded so too was a formal procedure. Informally, Compendium II of Copyright Office Practices spelled out a reconsideration proceeding, but it stopped at the Chief of the Examining Division. In the Copyright Reform Act of 1993, which did not pass, an appellate procedure would have been established. Even though the Act didn't pass, Ms. Peters instituted an interim one on May 4, 1995. The Franklin Pierce Law School has a very helpful database of Appeals Board decisions from October 30, 1995 to March 14, 2004, at this link. The decisions are worth reading.
One thing to bear in mind is the Office's interpretation of what a "final decision" is. For purposes of an APA review, one has to follow the appeals process. But for purposes of filing an infringement suit, the rejection from the line examiner suffices.