Koehler, head of state from 2004 to 2010, passed away in Berlin in the early hours, surrounded by his family, the Germany presidency announced.
President Frank-Walter Steinmeier led tributes to Koehler, saying that Germany had "lost a highly esteemed and extremely popular person who achieved great things –- for our country and in the world".
In a post on X, Chancellor Olaf Scholz hailed Koehler as a "committed politician who worked throughout his life for a fairer world".
An economist by training, he was the first German head of state who was not a career politician.
Before becoming president he was the head of the IMF in Washington from 2000 to 2004, and also held other roles in the civil service and banking.
He became Germany's head of state, a mostly ceremonial role, after being put forward by then opposition leader Angela Merkel, who went on to become chancellor.
Steinmeier said that when he was elected, the member of the centre-right CDU party was "virtually unknown".
Top-selling tabloid Bild memorably marked the news that he was seeking the presidency with the headline "Horst who?"
But the plain-speaking, unconventional politician quickly became popular in particular due to his criticism of big salaries earned by executives and the workings of the financial markets.
He was elected for a second term in 2009.
But in May the following year he stunned the country by abruptly resigning over remarks in an interview where he appeared to link Germany's military deployment in Afghanistan with defending economic interests.
The comments sparked controversy as there is still great sensitivity in Germany about overseas military missions due to painful memories of the Nazi era.
'Outstanding service'
In his speeches, Koehler frequently emphasised the importance of believing in the strength of Germany and in the energy and creativity of its people.
The married father of two promoted Germany as a "land of ideas" that would shape its own future and act as a force for good in the world.
When it came to foreign policy, Koehler had a strong focus on Africa and visited the continent many times.
"He never tired of pointing out the importance and potential of the African continent for global development," wrote Sven Behnke, the head of the ex-president's office, on Saturday.
"For him, engaging with Africa seemed not only wise and forward-looking, but also ethically imperative."
Before becoming president, he was a key economic official in the 1980s and '90s when chancellor Helmut Kohl was in power.
He played a central role in Germany's monetary unification in 1990 -- when the former East Germany adopted the West German mark.
He also helped prepare for the introduction of the euro.
He was born in 1943 to German parents in German-occupied Poland. He spent his early years living in refugee camps until his family settled in Ludwigsburg, in the state of Baden-Wuerttemberg.
After leaving the presidency, Koehler was the United Nations' envoy to Western Sahara from 2017 to 2019, leading UN efforts to end the decades-old conflict between Morocco and the Algerian-backed Polisario Front.
He was also a member of a panel set up by then UN secretary general Ban Ki-moon in 2012 to advise on the global development agenda beyond 2015.
Merkel, who was chancellor from 2005 to 2021, praised Koehler's "cheerful, optimistic and fearless" attitude.
He "has rendered outstanding services to Germany," she said in a statement.
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