How many Jewish refugee have been expelled out of Arab lands and when?

Here is a link to direct statistics on the Jewish Exodus from Arab and Muslim lands which provides a very clear and detailed answer (first result in Google if you copy-paste the title question, so this may be off topic and more for political show). About 800'000 Jews left Arab countries to settle in Israel between 1948-1973. Their departure was precipitated by a combination of rising religious tensions and incentives placed by Israel to encourage migration to the nascent state.

World War II and the events in the previous decades shattered a well-established and relatively peaceful co-existence between Jews and Muslims in the Middle-East.

During waves of persecution in Medieval Europe, many Jews found refuge in Muslim lands.[15] For instance, Jews expelled from the Iberian Peninsula were invited to settle in various parts of the Ottoman Empire, where they would often form a prosperous model minority of merchants acting as intermediaries for their Muslim rulers.

History of the Jews under Muslim rule, Wikipedia

During the period shortly before WWII, and the period immediately following it, a large increase in the influx of Jewish refugees from Europe combined with the devastated post-Ottoman economy and growing militias in the British Mandate Palestine created a hotbed for religious tension.

The majority of Arab Jews left the surrounding countries immediately following the establishment of the State of Israel in 1948, reflecting a sharp increase of the religious tension around that event, but also a success of Israeli policies discussed below. The declaration of independence was also the trigger for the worst anti-Jewish riots that targeted business' and individuals in which, combining all the events in several Arab nations, several hundred people died, more were injured and lost their property.

Waves of exiles from Arab Lands prior to 1947 however were relatively small and isolated, possibly indicating that religious tension present in areas with large recent immigrant populations had not yet spread to the countries where other long-established communities lived.

A portion of the total refugees left following economical incentives placed by Zionist organisations before the foundation of Israel, and later Israel. Israeli leaders were seeking to rapidly enlarge the Jewish population at that time for various strategic and demographic reasons.

Most notably, almost the entire Yemeni Jewish population was airlifted to Israel following a deal between Israel and Yemen during Operation Magic Carpet). This operation ended up being heavily criticised in Israel as it resulted in 850 deaths due to mismanagement. In perspective this number is about equivalent to the total number of Jews who died during riots in Arab countries in the mid-20th century (based on a rough addition of the numbers provided in the other answer).

This passage from Tudor Parfitt sums up all these reasons pretty well:

economic straits as their traditional role was whittled away, famine, disease, growing political persecution, and increased public hostility, often a desire to be reunited with family members, incitement and encouragement to leave from [Zionist agents who] played on their religious sensibilities, promises that their passage would be paid to Israel and that their material difficulties would be cared for by the Jewish state, a sense that the Land of Israel was a veritable Eldorado, a sense of history being fulfilled, a fear of missing the boat, a sense that living wretchedly as dhimmis in an Islamic state was no longer God-ordained, a sense that as a people they had been flayed by history long enough: all these played a role...Purely religious, messianic sentiment too, had its part but by and large this has been overemphasised.

Even before the founding of the State of Israel, anti-Jewish pogroms and government policies forced Jews to leave their homes and properties in Arab nations. The following is from a Jerusalem Post Supplement article called "The Forgotten Exodus," November 29, 2002, and from other sources as noted.

Iraq

By the summer of 1941, 180 to 200 Jews were murdered and 700 injured in pogroms in Baghdad. See also Levin, Itamar (2001). Locked Doors: The Seizure of Jewish Property in Arab Countries (2001 ed.), p. 6. Damage to Jewish property ran into millions of dollars, or $48 million in 2014 dollars. After 1948, the Iraqi government banned Jewish emigration, and hundreds of Jews were convicted for the crime of "Zionism," and sentenced to internal exile or fines up to $40,000 each. Tens of thousands slipped out of the country. By 1950, Iraq legalized emigration and pressured Jews to leave. Emigrants were permitted to take with them only $140 per adult; all remaining assets were confiscated by the Government. Thus came the end of a community dating back to the prophets Ezra and Nehemia that once had 190,000 and by 2000 only had 100 Jews. I have heard this confirmed by Iraqi Jewish refugees who are my neighbors.

Syria

The Syrian Jewish community, dating back to Biblical times, unraveled when pogroms erupted in Aleppo in 1947. All local synagogues were destroyed and 7000 of the town's 10,000 Jews fled. The government froze Jewish bank accounts and confiscated Jewish property. As the Syrian Jewish population dropped below 5,000, Syria enacted decrees banning Jews from being able to emigrate, sell their property, or work in governmental offices. In 1975, President Hafez al-Assad explained why he refused to allow Jewish emigration: "I cannot let them go, because if I let them go how can I stop the Soviet Union sending its Jews to Israel, where they will strengthen my enemy?" See Gilber, Lela, "Thank God There are Almost No Jews in Syria Now," National Review, Sept. 14, 2013. I know numerous Syrian-Jewish families who can attest to these acts; the most recent emigre left in the early 1980s.

Egypt

Egypt's government put physical and economic pressure on Jews to leave the country. In 1956-57, the government nationalized Jewish property, gaining assets totaling $2.5 billion. The Jewish community's population dropped from 75,000 to 200 by the year 2000, consisting by then mostly older people unable to travel.

Yemen

Before 1948, Yemenites suffered numerous attacks from their Arab neighbors and attempts to steal their property. Also, in 1922, Yemen enacted a law that required any Jewish orphan under 12 to be forcibly converted to Islam and removed from any contact with his Jewish family. Also, Yemenite laws limited Jews from occupying only the most demeaning of positions. After the 1948 war, however, mobs rampaged sending Jews fleeing for survival and forfeiting their property to the state. Some 50,000 Jews left Yemen in 1949 with the help of Israel's Operation Magic Carpet. Many of these Jews had never seen an airplane. Again, I have a close neighbor whose parents were among those refugees; she was born in Israel shortly after their flight.

Libya

Although the Libyan Jewish community dated back 2000 years, it took just a few years to be reduced from 60,000 to almost 100 Jews in 1969 and 20 by 1974. In November 1945, mass anti-Jewish violence resulted in the deaths of 120 Jews and the wounding of 500; 2000 were left homeless and all synagogues were burned.

Algeria

After Algeria achieved independence from France, the new government issued a variety of anti-Jewish decrees resulting in 160,000 Jews fleeing the country. All but one of its synagogues were seized and turned into mosques. By the 1990s there were 50 Jews left in the country.

Morocco

Morocco traditionally had the best relations with Jews due to good personal friendships between Morocco's King and certain Jewish professionals. Nevertheless, massacres in 1948 caused more than one out of seven of Morocco's 350,000 Jews to flee. Fewer than 2,500 remain today.

Tunisia

Prior to Nazi German-occupation in 1942, Tunisia had a thriving community of more than 100,000 Jews, about half of whom lived on the island of Djerba and used its 1900-year-old synagogue. In 2002, Al Quaida terrorists exploded a bomb at the synagogue killing 21 persons and wounding over 30. Today only 1,700 Jews live in Tunis and Djerba. Ibid.