This question is divided into two separate questions: a. The prohibition of Mefarek (extracting) / Sochet (squeezing)—since one desires to extract the tea from within the cotton. b. The prohibition of practicing medicine (Refuah) on Shabbat.

Regarding the prohibition of Mefarek, it is indeed forbidden (Shemirat Shabbat Kehilchatah, Ch. 32, Sec. 59), because we want the liquid—the tea that is inside the cotton. Regarding the prohibition of practicing medicine, the guiding principle is that any action whose medicinal nature is recognizable is rabbinically forbidden. However, the law is different regarding the eye, as the eye and the heart are interconnected; therefore, in cases of an eye inflammation, authorities are lenient regarding the placement of tea or ointment, after which the moisture in the eye itself will cause it to spread inside. Therefore, one may place tea in a manner that allows it to spread in the eye, without squeezing it from cotton or cloth.