There is a debate among the Halachic authorities as to whether one may pour hot water from an urn on Shabbat into a cup that has droplets of water inside it. It often happens that a person washes a teacup before using it, and some residual droplets of water are in the cup when he pours the hot water from the urn. Hacham Ben Sion Abba Shaul (Israel, 1923-1998) ruled that this is forbidden on Shabbat, and that one must ensure that the cup is completely dry before pouring hot water into it. Pouring hot water into a wet teacup effectively "cooks" the droplets in the cup, in violation of Shabbat. Hacham Ovadia Yosef, however, rules leniently in this regard, and allows pouring hot water from an urn into a wet teacup, after one has shaken out the main water from the cup. He offers several reasons for this ruling, including the fact that cooking through "Irui," pouring from the original utensil, may have a different status than cooking in the original utensil. Additionally, when hot water falls on top of cold water, it is not clear that it "cooks" the cold water; we may view the cold water as simply merging with the hot water, and not as being "cooked" as defined by Halacha.
According to all opinions, however, it would be forbidden to place a hot pot that one has just taken off the fire onto a wet counter. In this case, the aforementioned considerations do not apply. Here, the water is coming in direct contact with the utensil, rather than with hot water poured from the utensil, and this is not a case of water mixing with water, but rather of water touching a hot surface. One must therefore exercise care on Shabbat to ensure that the countertop is perfectly dry before placing a hot pot on it. Countertops are often wet as a result of water from the sink or spills, and if one places a hot pot directly on liquid on Shabbat, he will be in violation of the prohibition against cooking on Shabbat. One must therefore ensure that countertops are dry before placing hot utensils from the fire on the counters.
Summary: There is a debate among the authorities as to whether one must thoroughly dry a teacup before pouring hot water into it from an urn, and those who are lenient have on whom to rely. According to all opinions, however, one may not place a hot pot taken off the fire onto a wet countertop; one must ensure that the counter is perfectly dry before placing a hot pot onto it.