According to one view among the Rishonim (Medieval Halachic scholars), it is forbidden on Shabbat to pour hot water from its original utensil onto raw food or liquid. For example, it would be forbidden to take a plate of raw meat, place it under a hot water urn, and pour water from the urn directly onto the raw meat. According to this view, "Irui" pouring from a utensil onto food is equivalent to placing the food inside the utensil. Hot water poured from a Keli Rishon the original utensil in which it was heated is considered as "cooking" the food or liquid onto which it is poured, and is thus forbidden on Shabbat.
Therefore, if a person wants to make himself a hot drink on Shabbat, he must ensure that the teacup or mug is completely dry before he fills it with hot water from the urn. If one rinses a cup before filling it from the urn, there will, inevitably, be some drops of water on the bottom of the cup. If he fills the cup with hot water without drying it first, those drops will be "cooked" by the hot water from the urn, and he will thus be in violation of Shabbat. He must therefore ensure that the cup is completely dry before filling it from the urn.
However, if after one finishes his hot drink he decides to go for another, he does not have to dry his cup before refilling it from the urn. The residual drops in his cup from his first drink had already been cooked, and, according to one view among the authorities (the Rambam), one is allowed to cook on Shabbat liquid that had previously been cooked. Although we do not generally follow this view, we may take this opinion into account alongside other factors. Regarding the issue under discussion, there are some views among the Rishonim that pouring hot water does not have the Halachic effect of cooking. And even among those who maintain that pouring does have the effect of cooking, there are those who hold that this applies only to solid food, but not to liquids. Therefore, if the liquid had previously been cooked, we have enough of a basis to allow pouring hot water into the cup from the urn. It should be noted that we refer here only to a case where there are residual drops from the first drink in the cup. If there is a substantial amount of liquid in the cup, one should not add hot water directly from the urn.
By the same token, it is permissible to use on Shabbat a hot water urn with a gauge on the side that indicates the water level inside the urn. This gauge is filled with water, and when one pours some water from the urn, some of the water in the gauge leaves the gauge and mixes with the hot water inside the pot of the urn. One might have thought that it would be forbidden to use such an urn on Shabbat, since by pouring water from the urn, one indirectly causes cold water from the gauge to be heated. In truth, however, one may use this kind of urn on Shabbat, since the water in the gauge had previously been heated, and were dealing with a very small amount of water. Just as one may pour water from an urn onto a cup with residual drops if those drops had been previously cooked, similarly, it is permissible to use an urn with a gauge, since the water from the gauge that is reheated had previously been cooked.
Summary: Before filling a cup with hot water from an urn on Shabbat, he must ensure that the cup is completely dry, so that there are no drops of water inside the cup that will become cooked by the hot water from the urn. Therefore, if one rinses a cup before filling it from the urn, he must completely dry the inside of the cup before filling it. However, if one had a hot drink and now wishes to take some more hot water from the urn, he does not have to ensure that there are no residual drops from his first drink in the cup. Since those drops of water had been previously heated, one may fill the cup with hot water even with those drops inside the cup.
