The Shulhan Aruch (Orah Haim 326:2) writes that although one may not bathe his entire body on Shabbat, even with water that was heated before Shabbat, it is permissible to bathe with "Hameh Teverya" – the natural hot springs in Tiberias. Since these springs are naturally heated, the Rabbis did not enact a prohibition against bathing in this water.
In the next passage (326:3), the Shulhan Aruch cites two views as to whether this is allowed even in an indoor area, meaning, if a building was built around such a spring. According to the first view cited by the Shulhan Aruch, going into this structure is forbidden because it will, invariably, cause a person to sweat, and the Rabbis forbade intentionally inducing a sweat on Shabbat (as the Shulhan Aruch discusses later, 326:12). The second opinion, however, rules leniently in this regard, asserting that the prohibition against inducing perspiration on Shabbat applies only in situations where it is forbidden to bathe on Shabbat. Since bathing in "Hameh Teverya" is allowed on Shabbat, intentionally inducing a sweat in the area of "Hameh Teverya" is also allowed. Therefore, even indoors, where a person will perspire, bathing in these springs is permissible on Shabbat.
The practical application of this debate is the situation of a person who has the chills and wishes to relieve his chills by soaking his feet in hot water, which will induce perspiration. Although Halacha forbids bathing one’s entire body on Shabbat, it is permissible on Shabbat to wash a part of one’s body, such as soaking one’s feet. This, then, is a situation where a person seeks to induce a sweat in a circumstance where bathing is permitted. According to the second opinion cited above, therefore, it should be permissible to soak one’s feet in hot water with the specific intention of inducing perspiration. Hacham Ovadia Yosef writes that the Halacha follows the second opinion cited by the Shulhan Aruch, since the Shulhan Aruch cites both views with the expression "Yesh" ("there are those"), and when the Shulhan Aruch presents two different opinions with this formulation, Halacha follows the second opinion cited. Therefore, Hacham Ovadia Yosef ruled that it is permissible on Shabbat to bathe one’s feet in hot water – assuming, of course, that the water was heated before Shabbat – in order to induce a sweat.
Hacham Ovadia added that even if a person wishes to soak his feet for medicinal purposes – such as to alleviate his chills – this nevertheless does not fall under the category of forbidden medical procedures on Shabbat. Even healthy people soak their feet in hot water for warmth, and thus doing so does not overtly appear as a medical procedure. As such, it does not violate the prohibition of Refu’a (medical procedures) on Shabbat.
Summary: Although it is generally prohibited to intentionally induce perspiration on Shabbat, it is permissible on Shabbat to induce perspiration by soaking one’s feet in hot water that was heated before Shabbat, such as in the case of a person seeking relief from the chills.