I asked the proprietor of the Beis Medresh in question, and he said the main issue is קריע - tearing.
It is a discussion in Achronim (see here) exactly what the nature of tearing is that is forbidden. Following the view that it is separating two things that are glued or woven together (quoted in the פלא יועץ in the name of the Baal HaTanya and the Brisker) then a tea bag typically has a flap which is connected and closed. Opening that flap while avoiding separating it at the intended location would be a problem according to this view.
Interestingly, most people would probably follow the Mishna Brurah who argues on this view, but my understanding is that he forbids making tea in general due to issues of cooking.