I am a researcher from italy and I work in California as a postdoc, starting from this week. Most of my income comes from a fellowship issued from switzerland, which should not be taxed. However, a small stipend is paid also by the hosting institution to cover the high cost of life here (2000 $/month). On the basis of the U.S.-Italian Income Tax Treaty can I claim the exemption from taxes on this stipend? And in this case should I not fill any form or do it anyway and I am supposed to get the refun later?

I assume that you have J1 visa for Research.
You are non-resident for two year and your income in the U.S. for research is not subject to federal income tax, SS tax and Medicare tax.

Here is tax treaty between Italy and the U.S. from IRS Publication 901. Tax Treaties:
"A professor or teacher who is a resident of Italy on the date of arrival in the United States and who temporarily visits the United States to teach or conduct research at a university, college, school, or other educational institution, or at a medical facility primarily funded from government sources, is exempt from U.S. income tax for up to 2 years on pay from this teaching or research.

This exemption does not apply to income from research carried on mainly for the private benefit of any person rather than in the public interest. "

You will still file nonresident tax return Form 1040NR or 1040NR-EZ with Form 8843.

Actually, if he files the appropriate paperwork with the hosting institution in California to stop the withholding on any income taxes, he will need to file NO tax return at all.

He will just need to prepare the Form, sign the second page of the Form 8843, then mail it to the IRS center in Austin, Texas.