We can provide you with a source for what you heard, but we also must tell you that we do not agree with its application in the case that you describe.
The Rama[1] writes: “It is prohibited to pour from a kli that contains kosher fats into a lit candle cup that contains forbidden fat.” His source is a statement of the Mordechai[2] based on a mishna in Machshirin,[3] which states that when one pours a cold pure liquid into a hot impure liquid, the liquids are connected in the process, such that the contents of the pouring kli become impure. It is possible that a contemporary kashrut guide that you saw inferred from this ruling that one may not use an urn for both milchig and fleishig keilim. However, we will present you with the consensus of the poskim to the contrary and the logic behind that view.
The strongest argument against the Mordechai’s thesis is that one cannot compare the laws of transfer of impurity, which revolve around contact, to the laws of kashrut, which depend on imparting taste.[4] Indeed, given the fact that several Rishonim disagree with the Mordechai, we can understand why the Rama in the quote above concludes, “and if was already done, one need not be concerned.” The question in the poskim is thus whether one should avoid pouring, not what happens to the kli if one already poured.
Furthermore, your case is more lenient than that of the Rama in several ways. One is that the mishna says (according to the accepted opinion) that the stringency applies only when pouring from cold to hot, because the hot liquid on the bottom emits steam.[5] In contrast, in your case, one is pouring from hot into either hot or cold, and this situation is probably more lenient.[6]
There are further points of leniency as well. Rav S.Z. Auerbach is quoted[7] as saying that the Rama was speaking only about when forbidden foods are in one of the utensils, not about milk or meat. There are several leniencies with regard to taste from milk or meat food entering a pareve food and/or utensil.[8] Even if we were to rule that when hot water is poured from an urn onto milk, all the water becomes milchig, new water subsequently heated in that urn would be only b’chezkat chalavi (“milchig-leaning”). Regarding such water, there is a machloket between Ashkenazim (stringent) and Sephardim as to what extent and under what circumstances it can be consumed together with meat.[9] Therefore, it is logical to argue that the stringency of transfer through pouring into non-kosher need not be applied to more lenient areas. In our opinion, extending the stringency of not pouring from kosher to non-kosher to not pouring from a pareve utensil into actual milk or meat is reasonable but unnecessary.
In any event, it is certainly permitted to pour from a pareve urn into a milchig cup when the cup does not presently contain milk. After all, even Ashkenazim – who do not eat pareve food that was cooked in a fleishig or milchig utensil with food of the other type – agree that pareve food cooked with milchig or fleishig can be put while still hot into a kli of the other type.[10]
Based on the above analysis, we feel that it is perfectly acceptable to have one hot water kettle to be used with both milk and meat. One should keep in mind, however, that when a significant amount of hot steam reaches a kli from a food, it can change the kli’s status,[11] such that if an urn gets close enough to relatively thick steam of milk or meat, the type of issues we are discussing may exist.[12] Thus, in addition to making sure that the urn is not soiled by milchig and fleishig substances, it is proper to avoid putting foods that contain actual meat or milk too close to it, because of the steam that may arise from them.[13] However, one does not have to assume that the steam problem will certainly arise simply because he uses one urn with both types of utensils. Unless one makes a mistake, the kli will remain pareve. Further precautions beyond what we have mentioned are, in our opinion, unwarranted. However, you can check with your own rav to clarify his stance.