There are many opinions, including widely varied ones, on this matter. We will have to be satisfied with referencing the main issues and suggesting two of the mainstream approaches. There are other approaches, some more lenient (especially among Sephardic poskim) and some stricter. Our omission of these opinions is not meant to discount them.
If one pours from a kli rishon (the utensil in which the cooking took place) onto a food, it likely cooks at least part of the food, which is forbidden.[1] However, in general, we say that water in a kli sheini (a utensil into which hot food is transferred directly from a kli rishon) does not cook foods.[2] Thus, to prepare tea on Shabbat, it would seem to suffice to pour the water into the cup (usually a kli sheini) before putting in the tea bag.
However, in order to reconcile an apparently contradictory mishna, Tosafot[3] makes the following limitation on the leniency of kli sheini. Placing most uncooked foods into a kli sheini looks like cooking (michzi k’mevashel), and this is rabbinically forbidden. According to many authorities, tea leaves belong to the majority of foods to which this prohibition applies.[4]
There is another potential limitation on the leniency of kli sheini, which is likely to apply to our case. The gemara mentions two foods, salted fish and salt, that are or may be[5] considered cooked even when exposed to heat sources that usually are not sufficiently intense to use for cooking. The heat of a kli sheini is probably considered such a source. The question arises in the discussions of the Rishonim and Acharonim whether salted fish and salt are a closed list (and a limited problem) or two examples of a broad concern. The exact opinion of the Shulchan Aruch and Rama is somewhat unclear,[6] but the Mishna Berura[7] is stringent on the matter. Others note that tea leaves, which are very thin, are likely easily cooked.[8] So, most poskim looked for a different system to prepare tea.
One simple method, which enjoys the blessing of many poskim (including R. Moshe Feinstein[9]), is to pour the hot water from the first cup to another. Certainly, they reason, neither of the aforementioned stringencies should apply to a kli shlishi (a utensil where food was transferred from a kli sheini). However, others reason that if the water is hot enough to cook with, as it appears to get the job done with the tea, it doesn’t matter that it passed through one more utensil.[10]
Another issue is borer (selecting). A tea bag is made in a way that allows the tea flavor to seep out while the leaves stay put. As long as this takes place underwater there is no problem. However, when one lifts up the bag and waits for the last drops to fall into the cup (instead of on the table), he has used a specialized device to select the drops from the leaves (hence, borer).[11] Thus, we normally instruct those who use this method to remove the tea bag with a spoon and discard any tea droplets that materialize.
The “safest” mainstream approach (you don’t want to know what the Chazon Ish reportedly did!) is to prepare an essence before Shabbat by pouring boiling water from a kli rishon onto tea bags. Out of concern that not all the leaves are totally cooked, we do not reuse the leaves on Shabbat. One may pour the essence into hot water on Shabbat, as it has already been cooked, at least to the level of cooking it undergoes on Shabbat. Although many assume that cooked liquids may not be re-heated on Shabbat after cooling off,[12] they may be re-heated in a kli sheini.[13] Therefore, we put the water into the cup first and then pour in the essence.[14]