The questions surrounding adhesive name tags are those that surround the activites of sewing and tearing, which are activities already explicitly forbidden on Shabbat in the Mishnah. As in the case of all problematic activities on Shabbat, the concerns involve avoiding any kind of professional activity as well as not engaging in activities that are physically transformative in a significant way. Rambam Shabbat 10:11 is the first to make explicit that our concerns about sewing and tearing also apply to gluing and separating items.
In this case, there are probably three different stages of the process of using these sorts of tags that gave you pause, such that you asked the question: 1) Removing the sticker from its protective backing, 2) Applying the sticker to the piece of clothing, and 3) Removing the sticker from the clothing once the activity is over. Let's address them one at a time.
1) Removing the sticker from its protective backing
Rav Ovadiah Yosef, in Yehaveh Da'at 6:24, says this is not a problem, since the backing was only ever placed there with the intention of being removed by the consumer. [This is true, for him, of opening soda bottles and any other kind of packaging that is closed up only in order to open it later.] There are indeed some poskim who are stringent about this--see Mahazeh Eliyahu #70--claiming that the backing was intended to be on for a long time and separating it off is thus undoing what was at least a quasi-permanent connection. R. Ovadiah prefers being strict, if possible, and advocates removing the backing before Shabbat as a way of covering all bases. But in the current case, where you are describing an issue of higher cost and there is no feasible way to remove all the backings before Shabbat and still have the stickers work, he would agree that it is permissible to go ahead and remove the stickers on Shabbat. [Tzitz Eliezer 16:6 suggests another reason to be lenient here: One is not at all interested in the backing, such that it becomes like opening a disposable container to get at what is inside, which is permissible. He also offers a way of satisfying all views in this case: remove the backing and replace it before Shabbat, so that the new adhesion is obviously short-term and temporary.]
2) Placing the sticker on one's clothes
There is a long running debate among poskim as to whether the temporary stitching of two items together is forbidden or not. R. Yoel Halevi b. Yitzhak (Germany, 12th c.) and others permit this sort of temporary action, whereas R. Yeshayah of Trani (Italy, 12th-13th century) and others forbid. Beit Yosef seems to follow the lenient view in Siman 317, though recommends being strict in front of the unlettered halakhic masses, since they will have a hard time distinguishing between permitted and forbidden kinds of tearing and sewing. Indeed, Rema cites this approach as well in SA 317:7.
The case of a name tag is quite plainly a temporary adhesion and therefore is clearly permitted according to the lenient view above, which seems to be the core holding of the Shulhan Arukh. But given the concern about the uninformed masses, though one might be lenient in general, one might be hesitant in this sort of public situation (a community lunch), where people might get the wrong impression that connecting things with glue or other adhesives is simple not forbidden on Shabbat. But there is reason to be lenient here even according to those who are strict with temporary sewing and adhesions. Iggerot Moshe OH 2:84 (R. Moshe Feinstein, US, 20th c.) rules that it is permissible to fasten diapers and pendants with pins on Shabbat. He reasons that there is no issue of sewing at all, because sewing only applies in cases where two items have been made as one. Since the pin and the cloth are never truly joined into one entity, there is no issue. Similarly here, the tag and the clothing quite obviously always remain totally separate, and name tags are designed to be easily removable from clothing. [Hazon Ish OH 156, when he permits the use of safety pins in clothing, uses similar logic to explain why we should not be concerned about the halakhically unlettered masses when dealing with safety pins: this concern only applies to acts that look like the involve real stitches and real connections. This would seem to extend to something like an adhesive backing as well, where it is a fairly sui generis category not easily extended to other areas of practice.]
3) Removing the tag from one's clothes
On most clothes, there is no additional concern here above and beyond what we addressed above. Once the adhesion is not considered a Shabbat problem, neither is the removal. With some clothes (such as wool sweaters), however, little pieces of thread may tear off in the process of removal. Here, the relevant source is Yerushalmi Shabbat 7:2, 10a, where R. Huna rules that one whose clothing got entangled in thorns may remove them carefully in private, provided that he does not tear through his clothing. Rambam Shabbat 22:24 clarifies that even if the clothing rips in the process, there is no cause for concern, given that it was unintended. We are indeed often strict to forbid activities leading to unintentional consequences when we know for certain that the problematic consequence will occur, such that one might still see a problem for a person wearing the kind of sweater that will certainly tear somewhat when the name tag is removed. But R. Ovadiah Yosef in Yehaveh Da'at 6:24 permits removing an adhesive in a similar case, even if some tearing will certainly occur, since the maximal offense will be on the rabbinic plane, given that tearing is only a core biblical concern when it will be followed by some constructive/productive activity. Given that this is a case where the name tag will just be thrown out and no longer used for any other purpose, there is little reason to be concerned here. For those who might be uncomfortable, you can suggest that they may want to leave the name tag on until the end of Shabbat.
By Jaclyn Rubin and Rabbi Ethan Tucker