I keep all my vermouths (in the broad sense) in a refrigerator at 43 degrees F and all my vodka in the freezer. Some people keep their gin in the freezer also but I do not. There is no reason you couldn't store any 80 proof plus spirit in the freezer. With many cocktails though the water from the melting ice is an integral part of the drink.
Although most of the dry ice will sublimate it will add some carbonic acid to the drink, which may or may not affect the taste.