Pages have been ripped from at least 265 copies of the diary and other related books, they added.
It is not clear who is behind the vandalism. A US Jewish rights group has called for a police investigation.
Anne Frank’s diary was written during World War Two, while the teenager hid from the Nazis in occupied Amsterdam.
The book made her a symbol of the suffering of Jews during the war.
Japan has no history of Jewish settlement and no real history of anti-Semitism.
Toshihiro Obayashi, a library official in West Tokyo’s Suginsami area, said: “Each and every book which comes up under the index of Anne Frank has been damaged at our library.”
The Simon Wiesenthal Centre, a global Jewish human rights organization, said in a statement that it was shocked and concerned by the incidents, and called for the authorities to investigate.
“The geographic scope of these incidents strongly suggest an organised effort to denigrate the memory of the most famous of the 1.5 million Jewish children murdered by the Nazis in the World War Two Holocaust,” associate dean Abraham Cooper said.
“Anne Frank is studied and revered by millions of Japanese,” Abraham Cooper added.
“Only people imbued with bigotry and hatred would seek to destroy Anne’s historic words of courage, hope and love in the face of impending doom.”
The book was added to the UN Educational, Scientific and Cultural Organization’s (UNESCO) Memory of the World Register in 2009.
Anne Frank’s diary was translated into Japanese in December 1952 and topped the bestseller lists in 1953.
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