One must tovel (immerse in a mikve) various kinds of dishes before first use, and may not eat from his own dishes until they are toveled. Is it likewise forbidden to eat from someone else's untoveled dishes?
In Hilchos Tevilas Keilim by Rav Dov Cohen, he brings sources which permit eating using untoveled keilim in someone else's house, although, if memory serves, R. Moshe Feinstein forbade it. It should be noted, that according to Rav Yaakov Kamenetsky and others, one is not obligated to tovel glazed crockery at all, as noted here.
Rav Eliezer Melamed notes that according to the majority of poskim, it is forbidden for a guest to eat from untoveled keilim. Nevertheless there are several poskim who allow it, and he permits one to follow them, especially if not doing so will lead to a problem with 'darchei shalom'.
Yechave Da'as 4-44 based on the Bais Yosef – a person may dine in a restaurant or hotel where the dishes were not Toiveled. The reason is since the dishes were initially purchased for commercial purposes – to earn a profit – rather than for private use, therefore they do not have to be Toiveled. However a guest in a private home, where the dishes have not been Toiveled, may not eat off the dishes. Since the dishes were purchased for private use – and specifically, for food preparation – therefore they must be Toiveled.
However HaRav Shlomo Zalman Auerbach Zatzal in Minchas Shlomo 2-66 says that one is allowed to be Maikil and eat if the host does not specifically serve him food. In this scenario, the guest does not have an obligation to Toivel the dishes. However, the guest must remove the food from the plate with a Toiveled fork and spoon. If the host specifically offers the food, then the guest may not eat off the dishes because of Lifnei Iver. However by a restaurant, he says that one may not be lenient because of commercial use. Nevertheless, you may still eat, because you are not the the one who is obligated to Toivel the dishes and the restaurant would never allow you to remove the dishes from their premises. Therefore, you may eat off of the not Toiveled dishes.
According to some opinions, if the dishes do not belong to you, since you do not have a direct obligation to tovel them, you can eat off of the other person's (or restaurant's or hotel's) non-toveled dishes.
HaRav Zev Leff, noted posek and mora d'asra of Moshav Mattityahu, allows one to rely on these opinions when the host would be embarrassed or offended if you did not eat the kosher food off of his non-toveled dishes.
In any case, the rabbinic requirement to tovel dishes in your possession, which were made by a non-Jew, has NOTHING to do with kashrus.