Japan, unlike Europe, never really adopted Christianity. Thus a lot of the narrative about Jews never really took hold in Japan (namely, the part where they killed Jesus Christ), and there were no historical grievances about them. That being said, Japan was (and still is) xenophobic.

In Tokayer's controversial "The Fugu Plan," the case is made that several Japanese officers attempted to make a Jewish settlement region in Japan, with the express goal of attracting Jewish immigrants to make Japan stronger. They had read a copy of the book "Protocols of the Elders of Zion," which was a Russian book that pretended to be official Jewish documents regarding a plan to dominate the world, a book that eventually contributed to the rising anti-Semitism in Nazi Germany. However, partly because of the lack of the aforementioned historical narrative, and partly because of errors in translation, the Japanese officers that read this instead took the logical conclusion that "Since Jews tend to attract money and power, if we have all the Jews... we'll have all the money and power." However, I question the accuracy of this text, so take this with a grain of salt.

As far as actual, implemented, documented Japanese policies, a significant amount of Jews escaped to Japan via Russia (most notably assisted by Chiune Sugihara, the Japanese vice-consul to Lithuania), entering the port of Kobe, which already had a Jewish population, enroute to Shanghai, which had been declared an open city, meaning one did not need a visa to enter. They lived in what became known as the Shanghai Ghetto, which, while conditions were harsh (partly due to the lack of aid from Britain and the US after Pearl Harbor) which was ultimately still better off than that of the Chinese residents who lived in the same area. Most of the Jews survived the war and moved to the United States after the end of the war.

As for the Jewish community in Kobe, they were considered Japanese enough to be allowed to stay in Japan, where they were treated about as well as any foreign resident of Japan was treated. This was partly due to an extensive network of aid setup by the Jewish community in America and Britain, but also due to Japanese indebtedness to Jakob Schiff, a Jewish-American banker who had made a substantial wartime loan to Japan during the Russo-Japanese war via his bank, Kuhn, Loeb, and Co. (which eventually became a part of Lehman Brothers).

One of my acquaintances attended the Mirrer Yeshiva, a Jewish school in New York. Members of that school were among the Jews that had immigrated to Japan as part of Chiune Sugihara's efforts. From what I understand, they're one of the few such schools to survive the war in Europe.

I'm answering from what I remember of IB History for this bit, so take it with a grain of salt:

Mussolini had refused to have the Italians take part in the deportations, and as such there was no holocaust in Italy as in the rest of German controlled Europe. The Italian people were generally Catholic so there wasn't as large a Jewish population, there were still harsh anti-Semitic laws passed before the war by the fascist government.

Now from a more reputable source:

There was one instance where the Catholics in Italy helped out the Jews in the one ghetto that was fairly established, when the occupying Germans made demands of the Jews for their gold the Catholics arrived with gifts to the synagogue to donate and help them reach the goal the Germans had set.

Out of the ~50000 Jews in Italy at the time there were 8-9000 who were deported by the Nazis, with few returning. This was for the most part after the Germans had occupied Italy.

By another source:

An Italian-born Israeli historian earlier put forth the thesis that during the German occupation, “the Jews once more had an opportunity to experience the deep and courageous sympathy of the Italians, who did not hesitate to expose themselves to great peril to help the persecuted.” The U.S. Holocaust Memorial Museum’s characterization is similarly representative: “Simple gestures of human decency were the hallmark of Italian rescue efforts even by Italian police officials who were forced to cooperate with the deportation.”

So overall for Italy at least there were race laws, such as one that forbade intermarriage of Aryans and Non-Aryans, but the general Italian population was not in favour of the deportations, and thus had one of the smallest roles in the Holocaust of any Axis or Occupied nation (save for Denmark where the people were forewarned and it was a massive national effort to assist in a mass exodus to Sweden)

On the subject of Japan, their Axis membership led to very few additional policies, the Japanese wartime attitude towards Jews was one largely of indifference, they considered them like any other foreigner and they were treated as such.

In mid-March 1942, the Japanese policy towards the Jews was set out in a message broadcast from Tokyo to all diplomatic stations in the Far East. The message declared that the fundamental policy towards Jews, as set out in a Japanese Diet declaration in 1938, would be only partly modified to account for the Axis alliance. Jews would still be considered as any other group of foreigners, although the distinction of "Jewishness" would be based on race and culture. But this distinction applied only to stateless refugees - which meant German and Polish Jews. Any expulsion of Jews from Japanese-controlled territory was considered contrary to the stated Japanese national policy of the Common Brotherhood of Mankind (Hakko Ichiu - literally "8 roofs, 1 house"). Therefore, Tokyo's official policy was this: Jews holding citizenship of any country would be accorded treatment comparable to citizens of that country. Jews without citizenship would be considered stateless, in the same category as White Russian émigrés. This group of Jews would be under surveillance because of their "racial characteristics." Another category of Jews, those who could be considered "useful" to Japan because of their political or economic influence, would receive the same treatment that they received prior to the war. Source

There was of course a form of genocide carried out by the Japanese often called the "Japanese Holocaust" against Chinese, Korean, IndoChinese and Southeast Asian. This can be classified as a Democide.

Tl;Dr: Italian involvement in the Holocaust was greatly reduced by Italian morality, and participation was by the occupation, in Japan there was no Jewish Holocaust, though there were other large scale killings.