Making tea on Shabbos:
A. The general rule of Keli Rishon/Sheiyni/Shelishi:
It is Biblically forbidden to place uncooked foods into a Yad Soledes [110 Fahrenheit or 45 Celsius] Keli Rishon, Iruiy Keli Rishon, or very hot Keli Sheiyni, on Shabbos. Certain uncooked foods, however, such as spices and liquids, may be entered into a Keli Sheiyni which is not very hot. Other foods, however, may not be entered into a Keli Sheiyni. All uncooked foods may be poured on from an Iruiy Keli Sheiyni [and entered into a Keli Shelishi], with exception to very delicate foods which can cook with minimal heat, in which case it is Biblically forbidden to do so. The following will discuss the status of tea herbs and tea bags regarding if they may be entered into hot water on Shabbos:
B. Placing tea into hot water on Shabbos:
Making tea in a Keli Rishon and Iruiy Keli Rishon: It is Biblically forbidden to place uncooked tea herbs/bags into a Yad Soledes Keli Rishon, or to pour water from a Yad Soledes Keli Rishon onto it. One who did so, may not drink the tea, or give the tea to others to drink, on Shabbos.
Making tea in a Keli Sheiyni: Some Poskim rule that tea is considered a spice and may thus be placed into a Keli Sheiyni that is not very hot. Other Poskim, however, argue on this assertion, and rule that tea is not considered a spice, and contains a possible Biblical cooking prohibition to enter into a Keli Sheiyni. Practically, we do not enter tea into a Keli Sheiyni.
Making tea in a Iruiy Keli Sheiyni/Keli Shelishi: Some Poskim rule that tea herbs, and tea bags, are possibly considered delicate foods that only require a light heat for cooking, and it is thus Biblically forbidden to pour from a Yad Soledes Keli Sheiyni onto tea [or to place tea in a Yad Soledes Keli Shelishi]. According to this approach, tea may never be made on Shabbos with water of the temperature of Yad Soledes. Other Poskim, however, rule it is permitted to pour hot water from a Keli Sheiyni onto tea, and it is permitted to place tea in a Keli Shelishi. Practically, while many suspect for the former approach and are careful to make tea essence before Shabbos, or to cook the herbs before Shabbos, many others are lenient. The ruling of the Chabad Poskim is to be lenient in this matter.
It is forbidden to make tea using hot water of a Keli Rishon, Iruiy Keli Rishon, or by placing into a Keli Sheiyni. It, however, may be made in with Iruiy Keli Sheiyni, or Keli Shelishi, although some are careful not to make tea on Shabbos using hot water that is Yad Soledes, even in a Keli Shelishi.
 Admur 318:9;11;24;29
 Igros Moshe 4:74; Minchas Yitzchak Likutei Teshuvos 29; Rav SZ”A in Minchas Shlomo 1:91; SSH”K 1 footnote 3
 Admur 318:11 “Any food which has not been [fully] cooked before Shabbos [whether liquid or solid], one may not soak it on Shabbos in hot [liquidly] food that is Yad Soledes, even if it will not dissolve in there at all. If one [transgressed and] soaked it in a Keli Rishon then he is liable for cooking.”; Michaber 318:4
Definition of a Keli Rishon: A Keli Rishon is the original pot of food that was taken off the fire, or off any other heating surface [even if it’s content is no longer Yad Soledes-See D]. A Keli Rishon has the ability to cook foods as will be explained.
 Admur 318:19; Michaber 318:10; See Shach Y.D. 105:5
Definition: The pouring of the content of a Keli Rishon that is Yad Soledes onto a food or liquid. [Admur ibid]
 Implication of Admur 318:20; Tur 318; M”A 318:34; Levushei Serud ibid; Tzemach Tzedek Mishnayos 3:5; as explained in Ketzos Hashulchan 124 footnote 31; M”B 318:76; and 48; Chayeh Adam Shaar Mitzvas Haaretz 2:9; Rosh Yosef; See Shabbos Kehalacha 1:38 and 2:4
Other opinions: Some Poskim learn that the bath tub case is an exception and was only made forbidden due to a decree that one may come to heat up also in a Keli Rishon. Hence according to them a very hot Keli Sheiyni would have the status of a Keli Sheiyni. [Machatzis Hashekel 318:35, based on Rashi 42a; Toras Shabbos 318:27; See P”M 318 M”Z 18]
What is the definition of a very hot Keli Sheiyni which has the status of a Keli Rishon? If the Keli Sheiyni is much hotter than Yad Soledes Bo it retains the status of a Keli Rishon. [Ketzos Hashulchan 124 footnote 25] Some Poskim write it is the amount of heat that a person would scold his finger if he entered it into the water even for a moment. [Yad Nichvas Bo] [M”B 318:48 in name of Chayeh Adam; See Shevet Halevi 7:42]
 Admur 318:11-12; 19; M”B 318:65 and 34; Mishneh 42a; See Michaber 318:9-10 that only prohibits a Keli Rishon
 Admur 318:22; 318:12; Michaber 318:12-13; Mishneh 41a
 Admur 318:12; M”A 318:18; M”B 318:42; Some of these foods carry a Biblical cooking prohibition, while others are Rabbinically forbidden due to it appearing like cooking. [see Admur ibid]
 Admur 318:11; Michaber 318:4
 Peri Megadim 447 A”A 9 [See however Shabbos Kehalacha ibid in Biurim for other areas that the P”M is stringent]; M”B 318:47 based on P”M ibid quotes in a number of places; Ketzos Hashulchan 124 footnote 52; Nimukei Orach Chayim 318 regarding salt; Igros Moshe 4:74; Sheivet Haleivi 7:42; and many other Poskim; See Shabbos Kehalacha Vol. 1 p. 80-82
Other opinions: Some Poskim rule it is considered like a Keli Sheiyni itself, and hence all foods that are forbidden to be placed in a keli Sheiyni would likewise be forbidden in a Keli Shelishi. [Yereim p. 134; Shevisas Shabbos Mevashel 23 based on Chasam Sofer; Chazon Ish 51:17] Practically, the custom is like the lenient opinions which hold that it is allowed to place into it all foods which are allowed to be poured on from a Keli Sheiyni. However, since there are opinions which are stringent by a Keli Shelishi, therefore when applicable one should rather pour onto the foods from a Keli Sheiyni than to place them into a Keli Shelishi, being that the pouring of a Keli Sheiyni is explicitly permitted by all. [Shabbos Kehalacha ibid]
 Admur ibid “The same applies for any [food] that is hard which one did not [previously] cook and can become edible through soaking or washing it in hot water, then if one soaked it even in a Keli Sheiyni, or rinsed it even with the pouring of a Keli Sheiyni and it [then] became edible, one is liable for cooking.”; Rama 318:4
 Tzemach Tzedek Mishnayos 41
 See Admur 318:1 and 17; Rama 318:9;
 Tzemach Tzedek Mishnayos 41, as explained in Shabbos Kehalacha Vol. 1 p. 51-52; Rav Lavut in Haaros to 318:12; Nishmas Hashabbos 2:29; Maharam Shick 132; See also Or Letziyon 2:30-3
 M”B 318:39; Ketzos Hashulchan 124:7; Igros Moshe 4:74-18
 Shabbos Kehalacha Vol. 1 p. 51-52
 M”B 318:39; Aruch Hashulchan 318:28; Bris Olam Ofeh 23; Az Nidbaru 3:23-24; Piskeiy Teshuvos 318:36
 Peri Megadim 318 A”A 35; Implication of Tzemach Tzedek Mishnayos 41, as explained in Shabbos Kehalacha Vol. 1 p. 51-52; Implication of Maharam Shick 132, Rav Lavut in Haaros to 318:12; Ketzos Hashulchan 124 footnote 21; Minchas Baruch 12; Igros Moshe 4:74-15 [Keli Shelishi]; Or Letziyon 2:30-3; Divrei Shlomo 2:153 ;’Nishmas Hashabbos 2:29; Menuchas Ahava 2:10-39; Shabbos Kehalacha Vol. 1 p. 80-82
 The reason: As there is no reason to assume it is Kalei Habbishul any more than any other food. [Ketzos Hashulchanm ibid] The fact that it colors the water would occur even if the water was cold and is hence not relevant to Bishul. [Igros Moshe ibid]
 The tea essence and cooked herbs may be placed into a Keli Sheiyni, and if still warm, may even placed into a Keli Rishon if it was cooked on the fire from before Shabbos [as opposed to merely placed in hot water]. See M”B 318:31; Ketzos Hashulchan 124 footnote 23; However, see Chasam Sofer 74; Piskeiy Teshuvos 318:36
 See Tzemach Tzedek ibid; Ketzos Hashulchan ibid; Shabbos Kehalacha ibid