Absolutely! As strange as this sounds, if the temperature is cold enough, eating ice or snow will dehydrate you. Consuming snow or ice will drop your core temperature, which will work your body to continue heating it.
Notice how your body shakes when it is cold. This is Hypothermia Stage 1. It is shaking to raise the core temperature. If the core temperature continues to plummet, you will start to get confused and lack coordination (Hypothermia Stage 2). Hypothermia Stage 3 is when your skin is cold and turns blue, you tend to have irregular heart rhythms, weak pulse and enlarged eyes. Beyond this point you will hit Hypothermia Stage 4 (apparent death) and die of both dehydration from the energy exerted by your body and from hypothermia (Hypothermia Stage 5). You’ll know when you are in a Hypothermic Stage 2 when you stop shaking or the shaking slows.
It is best to put snow or ice in a pot, melt it down to a warmer temperature (close to body temperature or just boil it and let it cool enough), then drink it.