I am on a fully funded PhD Teaching Studentship (funded through the university department), which I believed was tax free. However, I recently received a tax code in the post and, when I rang the tax company, they were insistent that I would have to pay tax on it. Does anyone know if this is correct? Everything else I've seen implies that PhD's tend to be tax free. Thanks in advance for your help.

Check this, then ask your uni what status you are http://www.hmrc.gov.uk/manuals/eimanual/eim06271.htm

I've never heard of anyone having to pay tax on a studentship so don't think this would be the case. Maybe speak to your uni's finance dept to find out what's going on.