Bathing the entire body in warm water on Shabbat is normally prohibited by rabbinic decree. This decree applies even if the water was heated before Shabbat. But it does not apply to a woman immersing in a heated mikveh in order to fulfill the mitzvah of becoming tehorah.
The absolute minimum for becoming tehorah is one complete dunk, during which the entire body, including the hair, is submerged at one time. This is clearly permitted on Shabbat. Well-established customs, such as the Ashkenazic practice of immersing once prior to the blessing and once after, are also permitted. More than this should probably avoided, unless the water in the mikveh is cold. If, however, there is a clear custom in a particular community to immerse additional times, then the rabbi of that community should be consulted.
In addition, if a woman’s hair is not wet prior to immersion, then it has a tendency to float to the surface, which would invalidate the immersion. For this reason, women generally shower immediately before immersing to ensure that their hair is wet. Since this is not usually possible on Shabbat, many mikveh attendents suggest immersing one additional time prior to the above mentioned twice (for a total of three) to assure a proper immersion. As the additional dunk is also for the purpose of the mitzvah, it is reasonable to permit this as well.