Hello,
I arrived in Boston at the end of August and officially started work on Sept 2nd.
I am working at a hospital as a fellow on a H1B visa.
Am I entitled to tax exemption and, if so, for how long and how do I claim it?
Thank you very much for your help.

Because the H-1B visa is a work visa, you probably are NOT entitled to tax exemption.

However, I need to know the NATURE of the work you are doing under the fellowship before I give you the final advice.

I am an orthodontist doing clinical work and research in cleft lip and palate.

In my opinion, the clinical WORK over-rides the research portion of the work. The income you earn will be fully taxable.