You can open a special bank account for foreign citizens if your place of residence, including your domicile for tax purposes, is located outside of Italy. One disadvantage of doing this is that the bank fees are higher.
Banking in South Tyrol is cheaper for those who live and work in the province: all you need is a valid passport and an Italian tax number in order to open an account at one of the three local banking groups or one of the national banks that have a branch in South Tyrol. Conditions may vary from bank to bank, but the stamp duty (imposta di bollo or Stempelsteuer) is the same everywhere: every bank account in Italy requires the holder to pay € 34.20 per year.