Minimal, it would seem. Hype and temporary 'bragging rights' aside, the championship was politically (not to mention economically) insignificant compared to events which happened just before, during or shortly after, such as President Nixon's visit to Moscow in May 1972, SALT I / II, and the 1973 oil crisis. A period of US - Soviet détente was already underway and would continue until the late 1970s. Détente may have facilitated the Fischer - Spassky showdown, but the showdown had no discernible effect on détente, however big a news story it was at the time.
In the New Yorker article Game Theory: Spassky vs. Fischer revisited which reviews David Edmonds and John Eidinow's book Bobby Fischer Goes to War: How the Soviets Lost the Most Extraordinary Chess Match of All Time, professor and Pulitzer Prize for History winner Louis Menand notes that:
One possible reason for the world’s interest was the Cold War, and for
most of their book Edmonds and Eidinow play up the Cold War aspects of
the match. This makes it a little surprising when, at the end, they
discount the whole idea.
In the words of the book's authors (Edmonds and Eidinow),
...the championship took place in the high blossoming of
détente....Though almost all accounts of Fischer-Spassky couch the
match in geopolitical terms, they are, in this respect, curiously
misleading. The encounter might have been seen by the public and
written up by the press as a cold war showdown, but in the kremlin and
in the White House, East-West showdowns were not on the agenda.
According to the article, the authors argue that, on an official level, neither the Americans nor the Soviets were enamoured with their respective 'representatives':
American officials, on their side, regarded Fischer mainly with fear
and loathing. [Then National Security advisor Henry] Kissinger’s intervention seems to have been motivated by
personal interest in the game, rather than by grand strategy. The
State Department informed the American chargé d’affaires in Reykjavík
to spend no government resources on Fischer’s behalf, and the chargé’s
own deepest desire was to get Fischer off the island as quickly as
possible....
And, on the other side, Spassky was far from a typical Soviet-era
athlete. He was a patriot, but a Russian patriot. He hated the
Bolsheviks and had little respect for the Soviet system (though he was
careful to extract the rewards to which he believed his
accomplishments as a sportsman entitled him). It gave him pleasure to
ignore advice offered by Soviet officials, and in Iceland he made his
seconds and other handlers miserable with frustration by his
insistence on doing things his own way.
Further, there was apparently no evidence that either the Americans or the Soviets tried anything 'underhand'; it seems that neither side thought the stakes were particularly high in terms of the Cold War:
Edmonds and Eidinow speculate vigorously, but they can’t find any
proof that the K.G.B., or anyone on Fischer’s team, did anything
underhanded. They also conclude, somewhat reluctantly, that official
Soviet involvement in the match was not unusually intense, and that
the press coverage was entirely non-ideological. This was, they
properly note, a period of superpower détente.
As a consequence of losing,
Back in Moscow, Spassky and his team were subjected to a humiliating
postmortem, and Spassky’s travel privileges were suspended (a standard
Soviet response to failure in international competition).
Fischer, on the other hand,
was received in the U.S. as a national hero — Richard Nixon
sent him a congratulatory telegram, inviting him to the White House.
But this was short-lived and
After...a few grudging public appearances,... Fischer went off the radar
screen.
So, while the world may have been "hooked' for a while, only the game of chess seems to have been affected. The profile of chess rose in America, as it did in other countries. Fischer forfeited his title in 1975 and Anatoly Karpov dominated the chess scene for the next 10 years (1975-85).
If you are looking for a sporting event which did have an impact on political events, the best candidate would probably be the 1969 Football War between El Salvador and Honduras, but even here the football match in question was more a symptom of other, underlying, causes. Also of interest is the India–Pakistan cricket rivalry, but here the sport has generally served more to calm relations than inflame them (despite the intense rivalry).
