I one met a visiting Iranian academic just after I'd learned about the Shahnameh. I'd also read the opinion of a French scholar who thought its language was, for a modern Iranian, like Montaigne for a modern French. The Iranian woman told me that very few people in Iran actually read the book. It's very long, and hard to grasp for untrained readers. But most people know some of its stories and characters, because they are often mentioned in everyday life, and because the abridged prose books are widespread.
BTW, I don't know which editions are the most popular in Iran. Wikipedia says the Shahnameh was heavily modified and modernized up to the 14th century, when its most famous illustrated edition was created. The book most read today is probably not a scholar edition.
She makes a fair point. Reading and fully understanding Shahnameh is not straightforward. The difficulty does not primarily stem from drastic linguistic change, although the language has evolved and been somewhat simplified over time, but rather from the nature of Persian poetry itself, which is often deliberately layered and intricate *.
That said, Iranian students are introduced to selected passages and stories from Shahnameh throughout their schooling. Teachers typically devote considerable time to these texts, as the work is closely tied to cultural identity and a sense of historical pride.
* Persian, in particular, is often described as highly suited to poetic expression. Its flexible grammar and word order allow for a degree of intentional ambiguity, and this interpretive openness is frequently regarded as a mark of sophistication (difficult to master at a high-level for a layperson). A single ghazal by Hafez, for instance, can be read as a dialogue with God, a beloved man, or a beloved woman, with each interpretation leading to a different emotional and philosophical resonance. This multiplicity is the core part of the artistry.
Personally, I did not truly understand Hafez until I fell in love for the first time. My vocabulary and historical knowledge remained the same, yet my experience of the poetry changed completely. What shifted was something more inward and spiritual and only then did I begin to feel the full force of the verses.
For example, consider the following (unfortunately) translated lines:
O cupbearer, pass the cup around and offer it to me --
For love seemed easy at first, but then the difficulties began.