One of the Persian influences is Nuvroz. It is the Persian New Year and is celebrated in many Muslim countries. I know that Nizari Muslims in particular treat Nuvroz as one of their big religious holidays of the year.

Could you be a little more specific about what you're looking for here (time period? Are you referring to the pre-Islamic Persian Empire? Aspect of Islam influenced?). What immediately comes to my mind is the influence of Sassanid art on the art of the Umayyad and early Abbasid dynasties. The Sassanids' "portrait types" of their kings were especially influential to the Umayyads, who were searching for their own visual identity. Sassanid-style portrait types of muslim caliphs inhabit Umayyad coins. Sassanian influence can also be found in Umayyad depictions of the human figure-- particularly semi-nude females (both in fresco and sculpted form)-- at Khirbat al-Mafjar, Qasr al-Mshatta, and Qasr 'Amra, all in Jordan. For more information, see Eva Baer, "The Human Figure in Early Islamic Art: Some Preliminary Remarks." Muqarnas 16 (1999), pp. 32-41.

First, the term 'Persian' is incorrect and Eurocentric. Persian is an ethnic group in Iran, whereas there are over 10 other ethnic groups in this country who speak Iranic languages. The correct term is Iranian, as the name of the country has always been since the reunification by the Sassanid Empire.

Iranians did not alter the Islamic Orthodoxy up until Shi'a Islam was selected as the official religion of the country in 14th century by Safavid Dynasty. Up until that point Shi'a Islam was not very distinct from other established schools such as Hannafi School, but Iran has had been culturally separate from Arab countries for many centuries at that point, which in turn caused people to blend a lot of Iranian elements into Islam. One of them is "ta'ziye" which is known as maatam in India. It's a blend of Hellenistic aspects of Iranian culture with the Islamic culture, a Passion Show performed to remind people of Hussein ibn Ali's struggle with liberty and promotion of Shi'a Islam in the 8th century.

Another influence of the Iranian culture in Islam is evident in India and Northern China. Islam was first brought to these realms by invasion of Persianate Ghaznavid rulers and as such, Islam is blended with Persian culture in these places. For example the Muslim ritual of cleansing before the prayer is referred to as abdast whereas Arabs refer to it as wozu.

In "Two Centuries of Silence" by Abdulhossein Zarrinkub we read that in the first two centuries of Islamic dominance in Iran, during the reign of Ummayid Dynasty, Iran did play a large role in refining the Khalifate but it wasn't until an ex-Zoroastrian Iranian commander who changed his name to Abu-Muslim overthrew the Ummayid dynasty and installed the Abbassid Khalifate which Iranian culture came to shone in the Islamic world. Culture, science, architecture etc were all mainly Iranian. Baghdad was filled with Iranian scientists and artists who made an undeniable and lasting effect in the Islamic culture. Also it's worth noting that little Arab scientists exist and most of them, including Aviccena and Khawrazmi were Iranian.