Can someone wash their feet on Shabbos with warm water?
It is permitted to wash feet with warm water. It is only prohibited to wash the entire body on Shabbos.
This applies only to water heated before Shabbos, and not to water heated on Shabbos itself.
The question, however, is how the hot water is provided, and one must ensure that no Shabbos prohibition is involved in providing the hot water. This depends on the water system that one uses.
Sources: See Shulchan Aruch, Orach Chaim 326, Mishnah Berurah 1.