The main difference between a hotel and a hostel is that a hostel has dorm rooms (exclusively, or in addition to private rooms).
I spent a month in Iran in May 2014 and I did stay in some dorm rooms in Yadz, Shiraz and Isfahan. Still those places felt more like hotels as the majority of the rooms were private and they didn't have a kitchen for guests. Still most of these places were actually really nice and much better than the cheap hotels I stayed at in other towns and it was easy to meet other travelers.
So (as of May 2014), no there are no real backpackers in Iran, but some nice hotels with cheap dorm beds. In most towns you have to stay in hotels. You could also try couch-surfing. It is also possible that some local invite you to stay with them.