This is a very complex question due to the long history between the cultures. I will try my best to provide an answer that will suffice without typing a book.
Now, I'll avoid the Sunni/Shia part at first because that's not the primary issue and it's a bit touchy of a subject considring many Iranians and Arabs are both Sunni and Shia and I don't want somebody to put up their defenses while reading.
During the Persian empires, the Iranians never ventured much into "Arabistan" (land of the Arabs) excluding the coastal provinces such as Qatar and Bahrain. In fact, many times throughout the era the Shahs (kings, Iranian) were allied with various Bedouin tribes against the many enemies that 'Eranshahr' (Aryan Empire, the actual name - not Persia) faced. Never was the Persian culture or language forced on somebody because it was the culture of diversity and acceptance established by Cyrus the Great when he became 'Shahanshah' (King of Kings) ~550BC. Instead of destroying and desecrated the conquered, he rebuilt their temples and accepted their religion into the Empire. There is a reason our ancestors were in the Bible (Cyrus, Xerxes, 3 Wise Men). Even during the Sassanid Empire, the more centralized and 'harshest' of the Iranian empires still had respect and acceptance of the religions of the Empire. So then why do the Arabs invade and force their religion, directly and indrectly (Jizya taxes), on somebody that didn't do the same to them? As you can imagine this could be a strong reason. And these ancient cultures hold grudges for a very, very long time, to the point where being called an Arab or Turk in Iran is kind of an insult..
In 651 A.D. the last Pre-Islamic Iranian Empire fell to the Arab invasion, and it wasn't a kind one. When the Muslims sacked Ctesiphon/Tisfun (near Baghdad - also Persian meaning Gift from God) the capital of the empires, they desecrated the palace and shredded one of the largest Persian carpets in the world woven with precious gems and silks, burned the "Derafsh-e Kaviani" (a banner of high significance) as well as destroying Zoroastrian temples and libraries. To this day, you can still find Persian World Heritage Sites that still have the desecrations of the Arab invasions - an example being the stone carving of Mithradates on Bisotun mountain, Kermanshah, Iran. About 200 years later, Iranians were able to regain power from the Arabs thanks to a historically heroic figure by the name of Ya'qub Saffarid, of which he sent the Caliph a letter saying he has the Derafsh-e Kaviani with him (Impossible since the Arabs also shredded the flag; more of a taunt).
So now we have both religious and cultural destruction, both are more than enough reasons.
Now, as for what you said about "Farsi comes from Arabic and Iranian and Arabic cultures are very similar, so why so much hate between them..." I don't mean to be offensive or rude, but both of these are blatantly wrong. The Persian language has existed for a very long time completely seperate from Arabic. The Arabs didn't have written language until far after the Persians and other 'civilized' cultures of the time did, as the Arabs were generally nomadic in this era. (Not to say Iranians aren't nomadic by nature either, many were up until the Pahlavi Shahs of the 20th century). Probably what you are referring to is the "script" of the language, which wasn't adopted into Persian until "New Persian." In fact, prior to Islam, Middle Persian (a.k.a Pahlavi - no relation to the former Shahs) was written in Aramaic and translated into various other languages as to be understood by the various peoples of the Empire. (Notice the trend?) The only cultural similarities between the two races is of geographic and religious, however, Iranians actually are more related to Northern Indians (Aryavarta) as the Aryan race. Iranians and Arabs don't even appear the same - both have distinct differences in appearance. Iranians are more so white than dark skinned on average, my family included (people often think I'm either Greek or Italian or just a regular ol' white boy).
Now onto the Islam, which I'll keep brief because I myself am not Muslim. Iran didn't officially become Shia until the Safavid dynasty of Iran as more of a political move rather than a religious one. Not to say that they didn't believe in Shia Islam moreso than Sunni Islam, but there was obvious political motivation as it kept a religious boundary against the Ottomans and other Muslim countries. Shia Islam itself was introduced into Iran LONG before the Safavid dynasty made it the official religion.
Many Iranians to this day, even as Muslims, still believe that what the Arabs did ~1,400 years ago was the worst thing that has ever happened to Iranian history. More so than the Turkish (Seljuq) invasion or the Mongol (Ilkhanate/Timurid) invasions. As the Iranian empires of the era were the center of all culture, technology, economy, etc. and all were larger than the Roman Empire. After the Arabs conquered Iran, that part of the world hasn't recovered since. The other races/cultures/religions don't have the luxury of hating the Arabs like Iranians do because they were culturally absorbed and assimilated. Iran was able to Persianize and spread Persianates (Persian sultanates) Eastwards with Persian as the language instead of Arabic, such as the Mughal Empire --- spreading all the way to Ayyuthaya (Thailand).
----AS FOR THE BEST ANSWER:
The Arab governments do not like Iran because its a power-struggle for dominance in the Muslim world, which Iran is winning. Many proxy wars are fought amongst the minor 'insurgents' to gain the upperhand, which is why Iran is funding the terrorism. Most countries with influence and funds attempt this, including the USA. As for your statement about "reading history..." I may have to suggest that you read more in depth... No offense. Both Harun Al-Rashid and Salah ad-Din were Iranian or Iranian born, so these were horrifically poor examples. Harun al-Rashid was born in Rey (Modern day Tehran, Iran) and Salah ad-Din was Kurdish (part of the Iranian race).
I hope all of this answered your question and everyone elses, regardless of how late it came. As for the Anti-Arabism, I hope no Iranian here would consider genocide as it is very un-Iranian and our ancient ancestors would be ashamed of us.. As Aryan is synonymous with Noble and we must conduct ourselves as such.
I hope this was an educating read.
Thanks,
A proud Iranian-American
P.S. --- Please stop calling it Farsi everyone, it's Parsi. The Arabs couldn't pronounce 'P' so they subsituted it with an F. For accuracy, it's Parsi from Pars, not Fars.