My husband is from Israel and started a postdoc at MGH hospital on J-1 visa in Harvard, Boston. As I have heard in first 24 months the J-1 holders are free of all taxes. However, it was told to my husband that because of the treaty between the US and Israel, he have to pay the Federal and State taxes starting from now and not after 2 years. I think that there is a mistake, since no other postdocs in MGH pay taxes first 24 months. Can you help me?

This is from IRS publication 901.
An individual who is a resident of Israel on the date of arrival in the United States and who is temporarily in the United States primarily to teach or engage in research, or both, at a university or other recognized educational institution is exempt from U.S. income tax on income from the teaching or research for a maximum of 2 years from the date of arrival in the United States. The individual must have been invited to the United States for a period not expected to be longer than 2 years by the U.S. Government or a state or local government, or by a university or other recognized educational institution in the United States.
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. The exemption does not apply if, during the immediately preceding period, the benefits described in Article 24(1) of the treaty, pertaining to students, were claimed.

Does he receive regular wages or a scholarship? Even if he gets paid through MGH, the salary might be classified as a fellowship/scholarship.
He would need to fill out a form 8233.
Depending on the source of income he might have to pay income taxes for it if the source is a private organization (like HHMI for example)