I can only speak for Jews (though there is some information about general minorities as well), but here's what I've got in that regard:

The Persian Jewish diaspora generally considers its community's high point to have been the 1960s, during the regime of Shah Mohammad Reza Pahlavi. The Jewish community of Iran was considered to be one of the, if not the, richest Jewish communities in the world and generally were able to assimilate very easily into Iranian society at large, becoming largely secular. Jews had largely equal rights, and the Shah's campaign of Westernization and secularization served them well; the Shah also de facto recognized Israel, and Iran was distinct among the Muslim nations, many of which had expelled or suppressed their Jews following the establishment of Israel. Jews were only a quarter of a percent of the population (about 80,000), but were overrepresented in terms of literacy, university attendance, and presence in the medical field. They had full freedom of worship and many religious and cultural organizations. (Prior to the Pahlavi dynasty in 1925, Jews had lived in generally unfortunate conditions, though they had improved gradually over time with increased exposure to the Western world; it was in 1925 with the ascension of Reza Pahlavi that true improvement began, coming to a head with the White Revolution in 1963.) As Jews had been in Persia/Iran for millennia, they considered themselves completely Iranian and considered the land and country their home.

When the revolution came, reactions were initially complex. Many Jews were fearful of what the Islamic Revolution would bring, especially given what they had heard about Ayatollah Ruhollah Khomeini's opinions about Jews and Zionists; leading rabbis of the community organized a fast day to pray for the community's wellbeing. A number of Jews supported the revolution, some ideologically (specifically the antimonarchical Association of Jewish Iranian Intellectuals) and some, later on, pragmatically. In addition, the Jewish-run Sapir Hospital provided treatment for anti-monarchy protesters in 1978 who wanted to avoid government hospitals.

Upon Khomeini's ascension, it seemed as though many of the worst fears of the Jewish community had come true; Jews were demoted, along with other non-Shiite minorities, to second-class status legally. On a positive note, Jews, along with Assyrian/Chaldean/Armenian Christians and Zoroastrians, were given official minority status based on their historical dhimmi status, which at least made them better off than those, like Mandaeans, who were given no official status at all. For example, Jews and Zoroastrians were given one permanent member each in the Iranian Parliament, with Assyrian-Chaldean Christians also given one member and Armenian Christians given two (and those such as Mandaeans, or even Christians besides Assyrian/Chaldean/Armenian, given none); however, they were not permitted into any other high office in the Iranian government. Groups such as Baha'i, which were considered heretical, were forced to go underground, with members often pretending to be Zoroastrian. Minority communities, even official ones, were placed under increased scrutiny. Jews specifically could face difficulty finding employment if they were unwilling to work on the Sabbath. Sharia law's attitudes toward dhimmi meant that Jews and other minorities faced legal inequities- for example, compensatory damages paid to the family of a minority murder victim would be half that given to that of a Muslim victim.

The Jewish community didn't officially start to panic and leave, though, until May 9, 1979, soon after the revolution, when they received word that one of the political prisoners murdered by the new regime was a prominent community businessman, Habib Elghanian, who had been imprisoned, tried and executed in very quick succession for the crime of associations with Israel and Zionism, which were complete anathema to Khomeini. This greatly shook the community, as Elghanian had been widely known for his great patriotism and contribution to Iran; the community at large, which, as mentioned, had always felt Iranian, now seemed extremely vulnerable. When seven other Jews were executed for similar reasons in 1980 alone, that feeling only intensified.

There were, overall, two immediate responses. The first, done almost immediately after Elghanian's execution, was a journey by a leading rabbi (or hakham) in the Iranian Jewish community, Hakham Yedidya Shofet, as well as several other men, who immediately drove to Qom, the holy city where Khomeini now lived, to plead the community's case. They intended to demonstrate to him both the Jewish community's loyalty to Iran, painting it as a broader principle in Judaism that one must follow the laws of their host nation (dina demalchuta dina), as well as that Iranian Jews had no interest in or connection to Zionism or Zionists. When they met with Khomeini, they were incredibly relieved to hear him say that "we recognize our Jews as different than those Zionists" (some translations I've seen insert words like "godless" and "bloodsucking" to modify "Zionists"), comparing Iranian Jews to Moses and the Zionists to Pharaoh. That very day, this slogan was written on the walls of all of the Jewish religious centers in Iran, a reminder to the Iranian people of Khomeini's promise to them.

That said, the second response was to leave. 20-30,000 Jews left Iran from 1978-1980 alone, mostly to Israel and the US; additional tens of thousands joined them in the decades following. In the US, their primary communities are in the Beverly Hills area of Los Angeles, where they form a tens-of-thousands-strong part of the very large Iranian expat community, and Great Neck, New York, on the North Shore of Long Island, where they form the largest concentration of Iranians in the United States. The cumulative effect of their flight from Iran- which often came with great difficulty, as Jewish movement was initially restricted- meant that estimates of the Jewish population in Iran range from 20-25,000 to as low as 8-9,000.

The Jewish community that remained could live with relative freedom and religious autonomy, though with the limitations mentioned above; as long as they completely and outspokenly established themselves as Jews- specifically loyal Iranian Jews- and emphatically not as Zionists, life could technically continue in an altered form, though it could be a delicate and difficult balance, as though antisemitism was considered foreign to Iran, the lines between it and anti-Zionism could sometimes seem blurred. For example, just going in under the 20-year-rule wire in 1999, 13 Iranian Jews, including a rabbi, kosher butcher, and teenage boy, were imprisoned, under threat of execution, under suspicion of spying for the US and Israel- both of which nations condemned this action. (They were later released in groups.)

On the fate of Baha'is: They were more harshly regarded and treated due to the claims of the Baha'i Faith and the perceived challenge to Shi'ih Islam and the authority of the clergy posed by the Babi/Baha'i religions historically. [Significant segments of the Baha'i community also came from Jewish and Zoroastrian backgrounds.] Ayatollah Khomeini and certain other clerics had a deep-seated resentment and hatred towards Baha'is. In 1955, a well-known cleric began a campaign against the Baha'is that led to mob violence. In the 1960s, Khomeini declared Baha'is to be "unclean" and 'apostates." The clerics and a large segment of the Muslim community, therefore, viewed Baha'is as not eligible for any human rights or protections. Even to this day, a number of Iranian authorities and clerics deny that the Baha'i Faith is a religion and argue that it is a political group that serves foreign interests. See http://iranpresswatch.org/post/16302/ayatollah-khomeini-deviant-bahai-sect-apostates-and-unclean/ and the discussion further in a reply to this comment.

[The status and treatment of Baha'is has been widely studied and commented on by academics and in reports by "Western" news media; Amnesty International; the United Nations Human Rights Council and reports by the Special Rapporteur on Torture and on Extrajudicial, Summary or Arbitrary Executions ; European Union, Canadian, and United States official reports; and by Baha'i authorities and representatives. See https://en.wikipedia.org/wiki/Persecution_of_Bah%C3%A1%CA%BC%C3%ADs. See, also, https://www.bic.org/focus-areas/situation-iranian-bahais/current-situation ; http://iranpresswatch.org/ (a record of known and identify reports and instances of the persecution of Baha'is primarily in Iran); https://www.amnestyusa.org/iran-thumbs-its-nose-at-the-world/ ]

At the time of the Islamic Revolution in 1979, the Baha'i community in Iran was the largest religious minority with approximately 600,000 Baha'is. By 1990, approximately half of all Baha'is in Iran fled as refugees or emigrees to other countries.

There has been a long history of opposition to the Baha'i community and its predecessor Babi community in Iran, particularly among certain Islamic leaders and groups since 1844. The followers of the Bab, the first Messenger of the Baha'i Faith who declared Himself in 1844, included a number of Islamic scholars and clerics (over 400). The new religion quickly spread and majorities of some villages converted. However, the Bab condemned the corruption, arrogance, and backwardness of the Islamic clergy. The Islamic clergy denounced the Bab and His followers (called Babis) and began to actively persecute them. The Babis, able to defend themselves, rose up in a number of locations to oppose the persecution. Between 1844 and 1853, perhaps 20,000 or more Babis died and the Bab was executed in July 1850. (See https://reference.bahai.org/en/t/nz/DB/db-42.html for an account of the execution and the entire text for an account of the period from 1844 to 1853 translated into English and heavily annotated and footnotes with references on source material.)

Baha'u'llah was tortured, imprisoned in 1852, and then exiled to Baghdad in 1853 and finally exiled to the prison city of Akka, Palestine by the Ottoman Empire in 1868 due to the opposition. Even though Baha'u'llah strictly prohibited the use of violence or force (even for organized defense) and prohibited interference in governmental affairs and partisan politics, the memories of the Babi period and the desire of the Islamic clergy to stamp out the heresy continued and many conspiracy theories and false allegations were often leveled at the Baha'is during the life of Baha'u'llah.

"Later the Faith has been accused of being a creation of foreign powers: first Russia, via forged documents purporting to link the Bab to Russian agents During the period of the 19th-century so-called “Great Game” competition between Russia, Britain and the Ottomans for Central Asia, Baha’is were also often accused of being British agents. The effect of this narrative became strong enough that frequently to this day, one’s political opponents could be discredited by associating them with Baha’is ‒ thus, foreign influence, which was particularly hated by the clerical class, became associated with Baha’is, who were often interested in the modernization of the country. Baha’u’llah was exiled from Iran; eventually, at the request of Qajar Persia, He and His family were sent to the prison city of Akka in then-Ottoman Palestine. Baha’u’llah died in exile, and was buried there. The location of Baha’u’llah’s burial site became the center of the Baha’i world religion and its nascent administrative organs. Subsequently this area became part of the British protectorate of Palestine, and later still, part of the new State of Israel. A nonsensical constant of Iranian clerical opposition to the Baha’i Faith is that Baha’is are, in some never-explained manner, agents of Israel. " See http://iranpresswatch.org/post/16302/ayatollah-khomeini-deviant-bahai-sect-apostates-and-unclean/

Periods of persecution and opposition have occurred at various times and in various provinces (particularly in Yazd and Isfahan) throughout the 1900s. While the government of Iran and the Islamic clergy sometimes harassed and persecuted Baha'is prior to the Islamic Revolution, the Baha'i community in the 1900s was noted for its disproportionate advancement in education, support for women's equality, and employment as professions (particularly in the medical professions). Due to educational advancement and a more open outlook towards the rest of the world and modern science, Baha'is were generally more prosperous and some became quite wealthy as a result in the 1960s and 1970s. This bred further resentment and hostility. (Professor Abbas Amanat of Yale has written about this and the subsequent resentment and mistreatment of the Baha'is in his texts on modern Iranian history.)

With the Islamic Revolution,The nine members of the Baha'i National Spiritual Assembly were arrested in 1980 and are now believed to be dead. Additional and subsequent Baha'i leaders were similarly arrested. The Baha'i Faith's administrative institutions were eventually banned and all Baha'i activities forced underground. After a new election, eight of the nine new members of the Baha'i National Spiritual Assembly of Iran were arrested in December 1981 and then executed. Baha'is in government employment were generally terminated. Some Baha'i businesses were closely. Baha'i properties (centers, Holy Sites, and even cemeteries) were confiscated, damaged, destroyed,, and/or desecrated. A total of approximately 200 Baha'is have been killed or executed (some without acknowledgement by the State) and many more have been harassed, arrested, tortured, suffered losses of or damages to properties, and/or have lacked legal protections from harassment or harm by various hostile individuals and groups.

Due to international pressure, Iranian authorities and clerical authorities were limited in terms of executions and imprisonment and sought, instead, to strangle the Baha'i community economically and through the systematic denial of certain rights. In 1991, a secret government memo was leaked outlining a detailed plan to suppress the Baha'i Faith through the systematic denial of rights to Baha'is in Iran and international efforts to suppress the Bahai' Faith through various means. The Baha'i Faith was not recognized as a religion; Baha'is were sometimes branded as spies (for Israel, the United States, the United Kingdom) or apostates (which technically meant a sentence of death for Baha'is who converted from Islam or in general). Baha'i students were often dismissed or denied entry or diplomas from higher educational institutions. Efforts by Baha'is to create alternative means of obtaining higher education were later suppressed, with a number of students and teachers arrested and the efforts banned. Since Baha'is are not allowed to lie or dissimulate about their Faith, this prevented and has increasingly been used to deny Baha'is many rights. Even Baha'i marriages could be denied recognition and children of a Baha'i spouse could be removed and given to Islamic relatives or parents.

Baha'is were often denied redress in courts of law in terms of seeking any aid or compensation or complaints for mistreatment. Human rights representatives, attorneys representing Baha'is in court, and even government officials who complain about the mistreatment of Baha'is have also faced harassment , loss of office or licenses, and even imprisonment.

As a result of these measures, it is estimated that approximately half of all Baha'is in Iran fled Iran (mostly in the first ten years after the Revolution). Due to restrictions and limits on flights out of Iran, including requirements that persons identify their religion, many Baha'is in the first ten years from 1979 to 1989 fled through various routes to Kurdistan (Iraq) and Turkey or to Pakistan as refugees and have now resettled in various countries around the world. These efforts to flee Iran were quite dangerous and persons employed to aid in travel were known at times to abandon the Baha'is if detected or threatened and even in some cases to turn Baha'is in to government officials or forces.