I was discussing making a cup of tea on shabbat and yom tovim, my chavar said that you can not take the tea bag out of the cup (keli sheini of course). I have never heard this before and while it is still fresh I wanted to ask the sha’ala.
It is permitted to remove the teabag from the water. However, some write that one should preferably remove the teabag with a spoon rather than by hand.
Note that many are careful not to use tea bags on Shabbos out of concern for bishul; some are lenient to make tea in a keli shlishi.
The Shemiras Shabbos (3:58) writes that there might be a concern for borer in the liquid that falls from the teabag. This does not appear to involve an actual prohibition, since this is not considered an “act of separation” and it is only residual liquid flowing out from the bag – but he nonetheless recommends using a spoon to remove the bag rather than doing it by hand.