Is there a Hadith to this effect: "If a fasting person feels thirsty, they should engage in remembrance [dhikr], for it moistens the tongue and quenches their thirst"?

There is no Hadith that links the remembrance of Allah with thirst, and there is no Sharia-based connection between them. Perhaps the questioner confused it with the authentic Hadith: "Let your tongue remain moist with the remembrance of Allah," which means to persevere in remembrance, and it is more emphasized during fasting and other times, but it has no relation to thirst. Ibn al-Qayyim mentioned many benefits of remembrance, and he did not mention among them the alleviation of thirst.