Why is there always a tension between Saudi Arabia and Iran?
What is the root cause? What are the other causes?
Is this tension only between the regimes or also between general citizens?
How was this relation during the Shah's time?
The root cause is religious: Saudi Arabia is Sunni Muslim; Iran is Shia Muslim. A schism followed the death of the prophet in 632 and the conflicts over his succession that resulted. The relationship between the two groups has been tense to conflictual ever since, with periodic bouts of violence.
The two countries arguably are these two blocks' leaders in addition to being regional powers. Another contributing factor is that Saudi Arabia is a US ally, whereas Iran is a US enemy.
During the Shah's time:
The year of the Iranian Islamic Revolution was "one of great ecumentical discourse", and shared enthusiasm by both Shia and Sunni Islamists. [...] However, this harmony was short lived.
The reasons this harmony was short lived revolve around tensions related to Sunni discrimination and Sunni political groups being accused of Saudi support.
As to Saudi royals' attitude to Shia:
Relations between the Shia and the Wahhabis are inherently strained because the Wahhabis consider the rituals of the Shia to be the epitome of shirk, or polytheism.
This is indeed a regional power struggle. Despite representing two different branches of Islam, there have been continuous diplomatic relations between Iran and Saudi Arabia since 1922 Diplomatic relations were frail but were maintained under the Shah and the Islamic revolution. Diplomatic relations were broken in 2016. The current crisis started when a Saudi Shia cleric got the death penalty in 2016. This incident was a way of telling Shia Saudis that loyalty to the kingdom is paramount. At the moment two wars are being waged with these two countries as adversaries:Syria and Yemen. The areas around the main Saudi oil wells (Dhahran) has a Shia majority, and may be under Iranian influence. That is why these two regional powers are clashing, and why several Arab countries have put Qatar under an embargo, in an attempt to show their allegiance to Saudi Arabian supremacy in the Gulf. Iranians and Arabs do not hate each other more than protestants and catholics do, but religious relics are used by both parties for geopolitical gain.
Wahhabi government of Saudis (More or less like ISIL) consider Iranian Shia people as polytheist (see here). So the hatred is not unexpected.
Two country are regional powers and so are rivals.
during the Shah's time: each country had a dictator under the control of the US. Despite religious differences noted above, each one were playing own role under US control.
After Iran revolution, Ayatollah Khomeini proposed unity of Muslims. But there was a problem: he addressed Muslim people, not governments. This could be an alarm to dictators in the region; specially Saudis. Two rival country suffer two dictatorship and now one of them overthrow its dictator and, by vote, establishes Parliament, new constitution that even the leader should obey, ... ;a potential danger. On the other side, Saudis has significant role in creation of groups like Al-Qaeda and ISIL, which have hostility toward Iran.
As noted above, before Iran revolution, there was two dictatorship under US control. After Iran revolution there is one government under US control and the other government independent of US, even US enemy. So...
For Saudi, Iran and the US relation see here.
I am adding another angle to this discussion. Clash of civilizations. The catch here is, only Iran had a civilization.
Iran that is ancient Persia is a country with a great history, culture and language that precedes its islamisation. They are a proud people and rightly so.
Though forced to accept a foreign religion, Iran has always maintained its separate identity and rejected arabisation unlike the whole levant and berber states.
I guess ( I am not sure) that the Iranians deliberately became another branch of islam just to spite the Arabs.
Iran considers Arabs as barbarians and a lesser people and hence there would always be differences between these two groups.
One more reason is that Saudi Arabia thinks that Iran have plans with Israel to destabilize their country. They have some evidence that Iran have plans to empower the Shia Muslims in neighboring countries like Yemen to use them to destabilize Saudi Arabia and get access to holy cities like Makkah and Madina.