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Richard Nixon was a former President of the United States. After his tenure as President, he spent some time in Springfield.
Nixon later (after his time as President) received an honorary degree from Springfield A&M, conferred on him by the school mascot Sir Oinks-A-Lot. Nixon remained a supporter of both the school and the pig. When Sir Oinks-A-Lot was kidnapped as a prank by Springfield University students, Nixon was one of the pig's "powerful friends" who exerted influence on the Springfield University dean to expel the pranksters.
During the Presidents' Day Pageant at Springfield Elementary School, Bart clowned around by putting eyes and a nose on his behind to make it look like Nixon's face. He then mooned a group of girls and said, "I am not a butt!"
In a flashback scene, Homer heard Nixon's voice as he delivered his famous "Therefore, I shall resign the presidency" line.
A picture of Nixon, along with the other U.S. Presidents, was seen in a book that Superintendent Chalmers showed Bart, before Bart developed an interest in Theodore Roosevelt. Bart was only able to identify a few of the Presidents, and Nixon was not among them.
When Homer was put on trial for his soul by the Devil Flanders, Nixon was summoned to appear as a member of the Devil's hand-picked "Jury of the Damned." Nixon protested that he wasn't dead yet (and had just written an article for Redbook), but agreed to serve after the Devil reminded him that he owed the Devil a favor.
When Homer was young and found a lamp with a genie inside he asked his father if he could wish to be the world's greatest yo-yo champion; Abe responded to his son thinking it would not be a good idea due to the possibility of him performing for then-President Nixon and accidentally assassinating him, thus making Nixon's Vice President Spiro Agnew his successor and Homer proceeds to imagine this.
 Behind the Laughter
Nixon was the 37th President of the U.S., serving from 1969 to 1974. He also served as Vice President from 1953 to 1961. Nixon holds the historical distinctions of being the only President to resign from office and the only person to be elected twice to both the Presidency and Vice Presidency. In addition, Nixon was a candidate for President in the election of 1960 (which he lost to John F. Kennedy). Consequently, he was prominent on the American political scene for a long time, as alluded to by his appearing in the Duff Beer commercial with JFK.
Nixon resigned from office as a result of the Watergate scandal. When first accused of wrongdoing, he declared "I'm not a crook" in front of a crowd of journalists (referenced by Bart's "Nixon moon" prank). As evidence of wrongdoing mounted, however, Nixon realized that he was vulnerable to being removed from office via the impeachment process and resigned voluntarily.
After his time as President, Nixon became a prolific author and an elder statesman in the area of foreign affairs, being consulted by both Republican and Democratic successors to the Presidency. He died of a stroke on April 22, 1994.
 Historical notes and Simpsons role
Nixon's and Burns' golf game probably would have taken place while Nixon was still in office, or shortly after he resigned. Nixon formally resigned from the Presidency on August 9, 1974, after announcing his resignation in a televised speech the evening of August 8. Before that, he was dealing with the Watergate scandal, which led to his resignation. He would certainly have been concerned about going to jail, and quite possibly would have looked forlorn enough to move Burns to let him win at golf. On September 8, Nixon received a full pardon from his successor Gerald Ford, which eliminated the possibility of his going to jail. The same day he was pardoned, Nixon suffered a severe attack of phlebitis, from which he didn't fully recover—and which probably would have prevented him from playing golf—until early 1975. At any rate, for the remainder of 1974 after his pardon, Nixon wasn't in any condition to go golfing.
Nixon's visits to Springfield (for Whacking Day and to receive his honorary degree from Springfield A&M) would have to have taken place after his time as President. His hair is shown as entirely grey on those two occasions, and it wasn't all grey until several years after he was out of office.
Regarding the sign on the White House tour: In real life, Nixon was an avid bowler, and spent plenty of time in the White House's alley while he was President. He found that bowling sessions were easy to fit into his work schedule. However, it's unlikely that he ever had back-to-back 300 games.
Nixon was in fact still alive at the time of his appearances (as opposed to being mentioned) on The Simpsons, in "Homer Goes to College" and "Treehouse of Horror IV". He died just a few months after those two episodes first aired.
After Nixon's death, Matt Groening wrote a scathing article in the editorial column of Simpsons Comics 4 telling kids that Nixon was a crook and not to believe the sanitized eulogies on television that were ubiquitous at the time.