The main source is Igros Moshe OC 4:74 Bishul 15
For more details, see http://doseofhalacha.blogspot.co.uk/2014/02/making-tea-on-shabbos.html
Tosafos (Shabbos 40b) writes that once water has been poured from the kettle (kli rishon) into a kli sheni, the water begins to slowly cool down and so cannot be considered bishul. There are some foods, however, that are cooked so easily (kalei habishul), that one must not add water to them even in a kli sheni.
The Mishna (Shabbos 3:5) writes that one may place spices into water in a kli sheni, as they won’t get properly cooked. The Mishna Berura (318:39), however, writes that tea is not included and must not be made in a kli sheni. R’ Shlomo Zalman Auerbach (quoted in Shemiras Shabbos Kehilchasa 1:n152) explains that the spices and tea nowadays are finely ground and cook far more easily than the coarse spices typical of the Mishnaic era.
The Mishna Berura and Aruch Hashulchan (OC 318:28) don’t allow one to make tea in a kli shlishi (i.e. pouring the water into a kli sheni and then into another cup) though R’ Moshe Feinstein (Igros Moshe OC 4:74 Bishul 15) allows it as he holds that a kli shlishi cannot cook kalei habishul.
The Mishna Berura writes that it is ideal to prepare tea essence before Shabbos while others use instant tea. One preparing such tea should still use a kli sheni to pour into a kli shlishi (See Igros Moshe OC 4:74 Bishul 16).
Although tzoveya, dyeing, does not apply to food, it is best to place the tea in the cup before the water so as not to colour the water in the cup (Sha’ar Hatziyon 318:65).
Those relying on R’ Moshe must leave the teabag in their cup or remove it with a spoon as straining the teabag would be borer (Minchas Yitzchak 4:99:20).