Ok before I answer this I'm going to say that culture is a bit of a vague term. It could mean anything from social attitudes and values to art, and everything inbetween. To be safe I'm going to do a section on art and a section on the effects on Persian society. If I've completely missed the point or haven't covered an area your interested in feel free to ask follow ups.

Art:

In terms of art the Mongols had quite a lot of influence on Persian artistic styles. However this was not so much in terms of them introducing Mongol motifs into art, instead they brought in motifs from other areas they conquered and came into contact with. So in Persian paintings we see landscapes painted in Chinese styles. Another good example is the Universal History/Compendium of chronicles which was compiled by Rashid al-Din, a Persian serving as a government minister for the Mongols. This manuscript was rather beautifully illustrated, and we can see a wide variety of styles influencing the art such as Buddhist, Siennese Italian, Chinese and Byznantine iconography. If you look at this and this, both from 1314 copies of the work, you can see that the illustrations have some rather Chinese influences. These manuscripts would would have been illuminated in government workshops in Tabriz which had artists from across the Empire working in them.

Society:

We'd don't see much concrete influence from the Mongols here as fundamentally they were a very adoptive empire. So for example while with the Roman Empire you see the process of romanisation where they spread their cultural values and styles we do not see a similair process of Mongolisation. Instead the Mongol conquerors tended to adopt the culture of those they had conquered, although there is of course debate about whether this was for pragmatic reasons or genuine change, the rate of change, etc. In Persia this adoption of Persian culture manifested itself most obviously in the Ilkhan Ghazan converted to Islam. That said I can think of two interesting ways in which the Mongols did affect Persian society and scholarship.

The first is with regards to treatment of women. The Mongols on the whole were pretty flexible about giving women authority out on the steppe you couldn't isolate women in the house they were needed to do things. As a result there was increased political involvement of women in the Ilkhanate. Some good examples in Iran are Terkhan Khatun a woman who ruled Kirman as a Mongol vassal, although she also benefited from her dynasty's Central Asian background which meant they were also looser on female confinement. That said this freedom was likely confined to the elite sphere, the Mongols usually did not try and interfere with the lives of its everyday citizens and so these practices likely did not spread to wider society. Furthermore after the Mongols left this liberalism did not really last.

The second interesting area deals also with the Islamic world more generally. The Mongol conquests were incredibly traumatic events for the Islamic world, not just in terms of material damage but in terms of mental shock. The fact that these invaders were non-Muslim posed a big problem ot Islamic intellectual tradition. In Islamic though the world could be divided into two, the dar al-islam (the House of Islam) where Islam ruled and the dar al-harb (the House of Strife/War) where non-Muslims ruled. A popular idea was that the dar al-islam would slowly and inevitably encompass the world. Even if by then the idea of constant jihadi conquest had been pretty much abandoned the idea that the dar al-islam might retreat was incredibly shocking. So when a bunch of pagans rocked up and conquered huge swathes of Islamic lands, specifically the very important Persian areas, they reeled mentally as well as militarily. Thus as a society they attempted to find ways to rationalise the Mongol conquests/make them seem part of God's plan.

A relatively common claim was that the Mongols were simply God's diving punishment for the wickedness of the world. However for Persian scholars living under Mongol rule calling your potential patrons demons from hell was not a particularly productive thing to do. The first major Persian piece of historical writing from this period to deal with the Mongol invasions was Juvaini's History of the World Conqueror. As he was an important member of the Mongol administration in the area he of course had to be somewhat circumspect about the issues involved. He basically deals with the issue by side stepping it. He ends his history with the destruction of the Hashahin, a feared and hated Shia sect that carried out political assassinations, which is a great triumph for Sunni Islam that was dominant almost everywhere in the Islamic world. He also reflects that while the Mongols were not Muslims their presence meant Islam could now access new areas of the world that the empire connected them to. The next major piece of Persian historical writing is Rashid al-Din's Universal History which is an astonishing piece of research that had united an international body of scholars from across the empire and displayed unprecedented knowledge of China and India, and even some basic knowledge of Europe. His justification of the Mongol conquest was far easier as by the the Ilkhans had converted to Islam thus he no longer had to try and reconcile Muslims being ruled by semi-pagan Mongols. Thus he could argue that the Mongol Ilkhans fitted into the tradition of great Islamic rulers.