This is an excellent question, and a question still being debated to this day on Khomeini's ideological impact to the pre-1979 religious and political structure of the Arab world. However, a better way of phrasing this question would be to ask "to what extent did the ideology of the Iranian Revolution have on the Arab world and sectarian tensions," as the most important idea is to question how much did the Arab world (more specifically Sunnis Muslims and Sunni monarchies) do to suppress Khomeini's ideologies. Since there are countless actors and individuals affected by the Iranian Revolution and the answer of this question can only be answered fully in the length of a book, let us limit our case study to a country like Saudi Arabia before and after 1979.
From the eyes of al-Saud family in Saudi Arabia, Iran under the Pahlavi Shah had been a somewhat reliable diplomatic ally during the height of the Cold War in the 1960s and 1970s against the Soviet Union. The Saudi's view point held the belief that Iran is a barrier to prevent the Soviet's communist (specifically their atheist) expansion into the Middle East. The Shah, who had the backing of the U.S., also had common interest with Saudi Arabia, more notably wary against the rise of the Ba'ath party in Syria and Iraq.
From a religious point of view, the Sunnis in Saudi Arabia and the Shiites in Iran lived more or less peacefully. Despite the popular assumption that Sunni and Shia had been constantly at each others' throats, both religious sects were not hostile to each other before the Iranian Revolution. The Pahlavi dynasty may be in charge of the world's largest Shia population and the Saudi monarchy vice versa with a highly populated Sunni demographic, neither had made the effort to emphasize their religious differences. To the contrary, the two Islamic sects had little to no play in the role of Saudi politics with Iran until the rise of Khomeini in 1980 with his calls for an "Arab revolution" and politicizing Shiism in Mecca and Medina pilgrimages.
The Iranian Revolution struck Saudi Arabia with a bang. The deep connection of the Pahlavi dynasty and the Saudi monarchy had suddenly been shattered by a radical extremist, Ruhollah Khomeini. Khomeini looked not only to dismantle everything the Pahlavi dynasty established in the Shah's 25 year reign, but also the Sunni monarchies in Saudi Arabia and the Persian Gulf. A very similar comparison can be made between Khomeini and Lenin, as both had the vision of breaking up the old world order through revolutions, with either Lenin's communism or Khomeini's Islamic revolutions.
The response from Saudi Arabia took on many forms. One of the major ways Saudi Arabia fought to contain Khomeini's advances was to combat extremism with extremism. The Saudi monarch began to enforce stricter religious guidelines, while giving more power to the Sunni ultra-conservative sect, the Wahhabis, a religious sect that focused on restoring the "purity of Islam." Another way Saudi Arabia responded was to isolate Iran internationally and it was not hard to do at all. The United States, who also had stakes in the Shah, found themselves at odds with Khomeini, who was neither a pro-Soviet puppet nor a pro-Western leader. But Khomeini's comments on the U.S. and Israel made it clear that neither countries will have a peaceful relationship with each other. Thus the Islamic Republic of Iran found itself diplomatically isolated from the rest of its neighbors.
It was not hard for Saudi Arabia or its fellow neighboring Sunni states to turn diplomatically against Khomeini's Islamic Republic, as Khomeini easily made enemies on his own. However, Khomeini's ideological message put a strain on the religious and cultural views that Sunni and Shia Arabs lived in. Saudi Arabia saw the imminent threat of Khomeini's call for revolution across the Arab world and sought to quell any opposition in its country, specifically the majority of Arab Shias residing in Saudi Arabia's Eastern Province. The Saudi monarchy's response to combat Khomeini's ideologies, while extreme to some, set forth the precedent of Saudi's policy in the following decades.
Sources for further readings:
Al-Suwaidi, Jamal S. Iran and the Gulf A Search for Stability. Abu Dhabi, United Arab Emirates. The Emirates Center for Strategic Studies and Research. 1996.
Gause III, F. Gregory. Oil Monarchies: Domestic and Security Challenges in the Arab Gulf States. Council on Foreign Relations. New York. 1994.
Vassiliev, Alexei. The History of Saudi Arabia. New York University Press. 2000.
Wilson, Peter W. and Graham, Douglas F. Saudi Arabia, the Coming Storm. M. E. Sharpe, Inc. 1994.
Demeche’s comment is excellent and I wanted to add a response covering Bahrain, which was quite strongly affected by the Islamic Revolution. Bahrain is in an awkward geopolitical situation, having a majority Shia population and a Sunni monarchy. The country’s demographics in the 20th century are traditionally split as Sunni Arab, Shia Arab, Sunni Iranian and Shia Iranian. While there were some long-standing tensions between Shia and Sunni Arabs prior to 1979, many of these were by then nearly historic: the Shia had been serfs until 1923, while the Sunni Arabs were tribal lords. But I think that should be considered in the context of pre-modern relationships between Bedouin and settled people (Hadara), between urban (or urbane - the Sunni elite) and the rural (the Shia villagers). It just so happened that the Bedouin/Hadara split was also a Sunni/Shia split.
If you were a Shia born in 1923, your prospects would not have been very different from your Sunni counterparts. You had the opportunity to go to school (the first school for Shia boys opening in the late 20s); oil was discovered in 1932, and by the time you entered the work force in the late 1930s you’d have found work at in the oil industry. Arabs were treated equally in the largely-British run Bahrain Petroleum Company: it didn’t matter if you were Sunni or Shia, you had the same opportunities, and the same glass ceilings (higher managerial posts were reserved for British personnel). This provided the opportunity to break down boundaries between the two social groups.
In the 1950s, when you were in your early 30s, the first major political movement occurred in Bahrain, and a political party - the Committee of National Unity - was formed with a steering group of eight men, four Shia and four Sunni. They very consciously alternated their public meetings between Sunni and Shia mosques. The Committee’s leaders were arrested and deported in 1957 in the aftermath of anti-British riots, but for the next 20 years, political groups continued to operate which were focused anti-colonial, cross-sectarian and Marxist.
In 1971, Bahrain gained independence and briefly flirted with constitutional democracy. Not all Shia bought into the left-leaning opposition; many of them stuck to their traditional religious leaders. They were seen as being in opposition to each other as well as in opposition (or not outright allies) of the royal government. In elections for the 1974 parliament, the religious Shia bloc and socialist bloc formed a loose alliance (agreeing not to contest each other’s seats) and swept into the new assembly. But this was just a flirtation; in 1975 the Emir suspended parliament and dissolved the constitution. The socialists were the most harshly punished, while the religious bloc largely kept their heads down.
The result was that by 1979, the leftist, socialist and nationalist groups which had dominated your adult life - you would be nearly 60 by now - were all but extinguished, and a false political calm had been created. This rise and fall of the nationalists and socialists in Bahrain mirrored the collapse of Pan-Arabism that occurred over the 1970s.
Then, the 1979 revolution changed politics and society forever. In 1981, an arms stache was discovered, allegedly smuggled in by militants inspired by the revolution. This coup attempt spurred the creation of the Gulf Cooperation Council later that same year. With the Bahraini leftists destroyed in the 1970s, Shia revolutionary ideals created a new political framework for opposition. While a person born in the 1920s would have seen sectarian boundaries blur over the course of their adult life, people born a generation later lived through a re-drawing of those boundaries. Over the 1980s and 1990s, being Shia and being in opposition to the government came to merge together as a single identity in the public view.
In the 1990s, Bahrain witnessed an intifada that was dubbed a Shia uprising. This is despite the fact that the movement began cross-sectarian and remained throughout (and indeed built bridges between the religious Shia bloc and weaker leftist bloc, much as had happened in the ‘70’s). But there is no denying that the religious Shia bloc was the most dominant side, and while they were not tied politically to Iran, the 1979 revolution did spur these Shia opposition activists. On the government’s side, the state began taking increasingly discriminatory actions, choosing to exclude Shia from senior government positions and depriving Shia areas of the same kind of infrastructure investment as non-Shia areas. Though Shia Bahrainis found some kinship with their Iranian neighbours, the vast majority of them look to Iraq’s Shia clergy for spiritual leadership. Despite this, the government increasingly took to claiming that opposition was Shia and Iranian-backed.
The result is that the 1979 revolution served to spur both sectarian government policy and Shia-based opposition activity in Bahrain. The potential for this to happen already existed before the revolution: the defeat of the left-leaning opposition over the 1970s and the closure of public spaces (leaving only the mosque as the last public space standing) created a vacuum for opposition politics which the revolution helped fill.