No. If a liquid is bottled to be used as a mixer or an ingredient that is combined with an alcoholic beverage to create a mixed drink, and it is not intended to be consumed on its own, then it is not subject to bottler's tax.
Some examples of mixers include margarita mix, bloody mary mix, daiquiri mix, cosmopolitan mix, mudslide mix, simple syrup, mojito mix, demerara syrup, falernum, orgeat, grenadine, bitters, and sour mix.
Note that if a liquid is bottled to be consumed on its own, even if the liquid is purchased to mix with an alcoholic beverage, it will be subject to bottler's tax (e.g., cola, ginger ale, etc.).