About 200 firefighters battled the blaze after the building, which Adolf Hitler is believed to have visited more than a dozen times between the late 1930s and 1940, caught alight late on Friday.
Investigators are probing if a suspect detained close to the scene was connected to the fire, police said.
There was no immediate indication of a possible motive.
Built in 1938 as a guest house for Nazi officials and their guests, the building known as the Kehlsteinhaus, offered Hitler and his high-ranking supporters a bolt hole.
It is separate to Hitler’s nearby summer residence, known as the Berghof.
Now owned by the state of Bavaria, the Kehlsteinhaus was classified after the war as a historical monument recalling the Nazi period.
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The site is now a mountain inn and houses a restaurant.
The building now serves as a “cautionary reminder of the abysses of the inhumane Nazi dictatorship”, according to its website.
Contrary to popular belief, the Kehlsteinhaus was not a gift for Hitler’s 50th birthday in 1939, the inn’s website said.
The site, which is about 16 miles (26km) south of Salzburg, Austria, is known for its spectacular mountain views.
