Rum is made from sugar cane. On Passover, we avoid chametz, which is produced any time grains (wheat, barley, spelt, rye, or oats) ferment (see What Is Chametz?).

Since sugar cane is not a grain, it is theoretically very possible to make kosher-for-Passover rum. However, since rum is often made with yeast that is extracted from grain, only purchase or consume rum that is certified as kosher for Passover.

In addition, there are some (including Chabad) who traditionally avoid all alcohol on Passover (with the exception of wine).