For the most part rum is made from molasses, the byproduct of sugar production. So when we study the issue of the environmental impact of sugarcane production we need to keep in mind that molasses is the waste product of sugar production. Rum is recycling!
That said, we're studying not just sugar but sugarcane production so let's look at its impact. Most of this information comes from Sugar: A Bittersweet History by Elizabeth Abbott (2008). See the references page for more information.
In Cuba where they couldn’t purchase pesticides and fertilizers due to economic issues, they made their own version of organic sugar farming.
Sugar beets are a rotational crop so they don’t need much fertilizer or pesticide. It doesn’t cause much erosion or contamination.
The sugar industry ruined the Everglades. It was protected by President Harry Truman, but sugar planters drained it and plantations' phosphorous runoff hurt much of the topsoil.
Abbott writes about sugar's impact: “The World Wildlife Fun reports, cane has likely ‘caused a greater loss of biodiversity on the planet than any other single crop, due to its destruction of habitat to make way for plantations, its intensive use of water for irrigation, its heavy use of agricultural chemicals, and the polluted waste-water than is routinely discharged in the sugar production process.’”
On the other hand, “Although Brazilian cane production is notoriously destructive to the environment, cane-derived fuel is precisely the opposite. It is much cleaner than fossil fuels and contains no contaminants such as sulfur dioxide. It emits much less carbon dioxide and protects the climate by vastly reducing carbon emissions, hence reducing pollution. It is sustainable. It yield 8.3 ties as much energy as as that expended to make it and, as new cane varieties are developed, will yield even more. Even its by-products are valuable, and Brazilian mills process them into electricity for their own use and to sell to the national grid.... Cane-based ethanol is the Twenty-First Century’s miracle-in-waiting.”