You can consume most rums safely if you're following a gluten-free diet in almost every instance, rum is gluten-free.
The few exceptions to this rule include some flavored rums. If you enjoy flavored rums, you'll need to contact the manufacturer to determine if a particular product is gluten-free.
While pure rum is gluten-free, beware of pre-made drink mixes, such as those intended for piña coladas many of these contain gluten ingredients as flavoring.
Again, contact the manufacturer if you're in doubt, or make your own piña coladas from scratch using coconut cream and pineapple. Colleen Graham, About.com's Guide to Cocktails, has a simple but excellent piña colada recipe here just make sure your maraschino cherries are gluten-free, too.