To my knowledge:
Two people whom Obama intended to nominate or nominated to the Federal Reserve have been refused confirmation: Peter Diamond and Rich Clarida in 2010-2011.
Two people whom George W. Bush intended to nominate or nominated to the Federal Reserve have been refused confirmation: Larry Klane and Randy Kroszner in 2008.
Two people whom Bill Clinton intended to nominate or nominated to the Federal Reserve have been refused confirmation: Alicia Munnell and Felix Rohatyn in
Before then… to my knowledge, at least, you have to go back to 1914 and President Woodrow Wilson and the first batch of nominations to get any other examples: Thomas Jones, Richard Olney, and Harry Wheeler.
Am I wrong?
Roger Johnson, "Historical Beginnings: The Federal Reserve" http://www.bos.frb.org/about/pubs/begin.pdf:
On May 4  the President [Wilson] sent his five nominations to the Senate. They were: Richard Olney, conservative Boston lawyer and Secretary of State under Grover Cleveland 20 years earlier; Harry A. Wheeler, Chicago businessman and former president of the United States Chamber of Commerce; Paul M. Warburg, partner in the Wall Street in- vestment firm of Kuhn, Loeb & Company, and an opponent of the Federal Reserve bill while it was before Congress; Adolph C. Miller, a former professor of economics at the University of California; and William P. G. Harding, president of the First National Bank of Birmingham, Alabama, and a champion of his own city as the site for a Federal Reserve bank.
Almost as soon as Wilson named his choices he faced embarrassment. Olney was probably the most prominent of the nominees, but his stewardship of the State Department had been filled with controversy, and, citing his advanced age as the reason, he declined the appointment. Wheeler also turned down the offer.
The President’s embarrassment soon turned into a nasty political confrontation with the Senate. While Wilson’s selections proved very popular among America’s banking leaders, the President’s natural political allies—the progressives—were deeply and bitterly disappointed. Within Wilson’s official family Secretary McAdoo strongly advocated the appointment of a Board that would work with him to break what he considered to be Wall Street’s control over the nation’s credit. The President rejected McAdoo’s argument in favor of the position of Colonel Edward M. House, Wilson’s most important adviser. House advocated the selection of men who would win the confidence and cooperation of the bank- ing community, and the President gave him a free hand to consult widely among conservatives and among the banking leadership for suggestions.
The progressives were appalled by the nominations, and the pleasure expressed by bankers and conservatives only deepened their suspicions. After Olney and Wheeler declined appointment, Wilson, on June 15, named in their place Charles S. Hamlin, a Democrat from Boston, and Thomas D. Jones, a businessman from Chicago. These replacements, particularly Jones, only angered the progressives further. Led by Senator James Reed of Missouri, the progressives directed more of their fire at Warburg and Jones. Warburg was suspect because he represented a prominent Wall Street investment house and because he had been a strong champion of that bete noir of the progressives, the Aldrich plan. Jones was suspect because he was a director of the International Harvester Company, a trust that was universally hated by the middle western farmers and progressives, and which was under both state and federal indictment in 1914 as an illegal business combination in restraint of trade. Wilson was particularly embarrassed and embittered by the opposition to Jones, for the latter was an old friend who had sided with him during his controversies as president of Princeton University and who had contributed large sums of money to his presidential campaign of 1912. Moreover, Jones had reluctantly accepted the appointment only after Wilson had appealed to him on the basis of their friendship.
President Wilson decided to fight vigorously for the Senate confirmation of his five choices, and he came out with particular force for his old friend Jones. He argued that Jones, as a director of International Harvester, had been working to end the activities which had brought that company under indictment. In July, Jones testified before the Senate Banking Committee, which was holding hearings on the President’s five nominations, and he weakened his own case by showing more sympathy with the policies of International Harvester than Wilson had suggested was the case. A few days later the committee voted, seven to four, to disapprove Jones’s nomination. Infuriated, Wilson determined to carry his fight for his friend’s confirmation to the Senate floor. Despite very heavy Administration pressure, a number of Democratic senators normally aligned with Wilson refused to accept Jones. The President, seeing that his friend could not prevail in a Senate vote, asked him to withdraw his nomination. Jones, who had not been eager to serve on the Federal Reserve Board in the first place, gladly complied. This was Wilson’s first defeat at the hands of either house of Congress.
As a replacement for Jones, the President nominated Frederic A. Delano, president of the Monon Railroad, and he was easily confirmed by the Senate.
Meanwhile the Senate Banking Committee had also requested Warburg to appear before it. Warburg’s pride was so wounded by this request—he seemed to feel that he was being asked to appear at an inquisition—that he requested the President to withdraw his nomination. Wilson refused to do so and pleaded with Warburg to appear before the committee as Jones had done. Senator Hitchcock assured Warburg that he would be treated kindly. In early August, Warburg finally consented to testify, and he was promptly approved by the committee and confirmed by the Senate. Apparently the defeat of Jones, and Warburg’s ultimate appearance before the committee, was victory enough for the progressives, for they made no serious attempt to block confirmation of Wilson’s three other selections. Perhaps most significantly, Wilson’s appointments to the Federal Reserve Board were very welcome to the banking com- munity, and they indicated that the President wished to inaugurate the Federal Reserve System in cooperation with the financial community of the nation.
On August 10, 1914, the Federal Reserve Board was officially sworn into office, with Charles S. Hamlin designated Governor (i.e., Chairman), and Frederic A. Delano, Vice Governor, and it took over the work that had been started by the Reserve Bank Organization Committee…