Is wearing hijab mandatory for Iranian women?

Yes, it is mandatory. However, if you're a foreigner, and your physical appearance attests to that (e.g you're not a black-haired Italian woman who looks just like Iranians!), nobody will really bug you if you don't know how to properly employ that or your scarf drops from time to time. Even if the morals police sees you, they are most likely going to give you a warning and ask you nicely (with tutorials on) how to properly use hijab.