Answer: The Mishna (Beitza 28a) teaches that one mustn’t measure things on Shabbos or Yom Tov. Tosafos (Shabbos 126b) explains that doing so is uvdin dechol, mundane, weekday activity, while Rambam (Shabbos 23:12) implies that it may lead to writing (See Shulchan Aruch OC 323:1). The Shulchan Aruch (OC 306:7) writes that one may measure, however, for mitzva and health purposes (See Shabbos 157b).
Thus, R’ Moshe Feinstein (Igros Moshe 1:128), R’ Yitzchak Yaakov Weiss (Minchas Yitzchak 3:142; 7:22; 10:31:8), R’ Eliezer Waldenberg (Tzitz Eliezer 3:10; 14:30), R’ Yehoshua Neuwirth (Shemiras Shabbos Kehilchasa 40:2) and R’ Ovadia Yosef (Yabia Omer OC 9:108:156; Yechave Daas 4:29) write that one may use a (glass) thermometer on Shabbos.
There is a machlokes, however, as to whether one can use a strip thermometer which changes colour according to the temperature. R’ Neuwirth writes that R’ Shlomo Zalman Auerbach only allowed one to use one if they can read the numbers or symbols beforehand Otherwise, one transgresses the melacha of koseiv, writing (See Minchas Yitzchak). R’ Waldenberg, however, writes that while glass thermometers are preferable, one may use such thermometers if necessary. R’ Ovadia Yosef, however, maintains that there is no issue whatsoever in using them (See Nishmas Avraham OC 306:7).
In conclusion, one may use a glass (alcohol) thermometer on Shabbos even if one isn’t particularly unwell. If necessary, one may even use a strip thermometer.