Before a woman immerses in a Mikveh, she is required to clean her teeth to ensure to remove all food particles, as the presence of such particles would constitute a "Hasisa" ("interruption") and disqualify the immersion. This is the ruling of the Shulhan Aruch (Yoreh De’a 198:24). The Shulhan Aruch adds that because of this Halacha, the custom developed that women refrain from eating meat – including both red meat and poultry – on the day of their immersion. As meat is prone to becoming wedged in between the teeth, women adopted the practice of refraining from meat on the day of the immersion to help ensure that no food particles would be present in between the teeth at the time of immersion. The Shulhan Aruch approves of this practice, describing it as a "Minhag Yafeh" ("a nice custom"). This custom is also mentioned in the Bet Yosef (listen to audio recording for precise citation), and was documented already by Rabbenu Yeruham (early 14th century), who lent his approval to the custom.
The Taz (Rav David Siegel, Poland, 1586-1667) writes (in Se’if Katan 25) that if a woman will be immersing on Mosa’eh Shabbat or Mosa’eh Yom Tob, then she may eat chicken and meat on that day. This custom is not taken so far as to require refraining from meat on Shabbat and Yom Tob. However, a woman in this case must take special care to ensure to clean her teeth properly so no meat particles are wedged in her teeth at the time of immersion. This conclusion also emerges from the Bet Yosef (199; listen to audio recording for precise citation), who cites the ruling of the Samag and the Sefer Ha’teruma that if Yom Tob falls on Sunday and a woman will be immersing on Sunday night, she must take special care to clean her teeth from the meat eaten over the previous two days. This ruling is codified in the Shulhan Aruch (199:6). The clear implication of this ruling is that a woman immersing on Mosa’eh Shabbat or Mosa’eh Yom Tob is permitted to eat meat that day, as long as she ensures to clean her teeth properly before immersion. This is the ruling of Hacham Ovadia Yosef, in Taharat Ha’bayit (vol. 3, p. 124).
It should be noted that this custom does not apply to meat soup, which obviously does not get wedged into teeth. Patties made from ground meat, however, are included in this Halacha and should not be eaten on the day of immersion.
Summary: It is customary for women to refrain from eating meat and chicken on the day of their immersion in a Mikveh. If she immerses on Mosa’eh Shabbat or Mosa’eh Yom Tob, she may eat meat that day, but she must take special care to ensure to remove all food particles from her teeth before immersion.