I'm an Italian citizen. Next Sunday I'm going to Austria. I have a national ID card but no passport. Recently Austria suspended Schengen. So can I enter Austria?
The purpose of the Schengen Treaty is to let EU/EEA citizens pass internal EU borders without having to wait in line to have their papers checked. This greatly simplifies cross-border travel and commerce since there are no delays. Earlier treaties allowed EU citizens to pass those borders without visa or passports, just with their national ID cards. Those are still in force.
Travelers still have to carry their ID cards under Schengen rules, and they must be prepared to show them during random spot checks by border authorities. The way Schengen is supposed to operate, such random spot checks are truly random and infrequent.
To my knowledge, no country has suspended the Schengen common visa policy or passport-free travel for EU/EEA citizens. Due to the refugee situation, some nations make the checks less random and much more frequent. While this greatly disrupts travel, it does not change who can enter and with which papers.
In Austria you are not required legally to have/carry any passport/id. You are, on the other hand, required to carry a passport/id when crossing the border.
Also, you are required to be able to identify yourself with appropriate documents. So what this means in practice is, that if the police stops you and you don't have any ID, you will have to make them believe you that you are an EU citizen. Worst thing that can happen is that they take you to the station and hold you there until they are sure you are an Italian citizen.
Legally, you need to carry a EU ID card or a passport when crossing the border. In practice, if you have any official document like a drivers licence, this will usually be enough if the police stops you, even if you wouldn't be strictly allowed to cross the border.
Your national ID card allows you to cross the border and you won't have any problems with it.
I haven't crossed the border between Austria and Italy since the start of the refugee situation, but I crossed the German/Austrian border a few times with only my driving license albeit the more stricter German law concerning identification. The police probably has other problems and won't bother spending time with someone who is clearly an European citizen.
Some information about this can be found here