Why is Rum not affected in the same way by distillation as other alcohols like Vodka? Is this true:


  Distilled alcohol such as gin, vodka, scotch, whisky and rye are
  produced from fermentation and distillation of wheat, barely or rye.
  The distillation process separates the alcohol from the gluten
  proteins, producing an extracted product that is gluten free. Despite
  being manufactured from grains that contain gluten, the final product
  contains NO gluten."
  
  Rum is distilled from sugar cane and is high in FODMAPs, so avoid if you can. However, it is a gluten-free option safe for those with
  celiac disease.


Why is vodka able to have it's gluten removed after distillation but when rum is distilled the sugar(s) [FODMAPs] is still an issue?

Quoted source: Alcohol That Doesn’t Make Your Belly Ache
There is nothing different about the distillation of rum. Just like with whiskey, brandy, vodka, etc. no sugar comes off the still. 

What does happen a lot in the rum industry is the post distillation addition of rum, as referenced in Eric's answer (the author at Alcademics - Camper English - has done a lot of research on sugar in rums). 

Sugar can have gluten contamination if it is produced near grain processing. Depending on the rum and where the producer sources their sugar, if sugar is added, there is a slight chance of this happening. 

If you want to avoid any chance of additive sugar, you can target brands that don't use additive sugar like Foursquare from Barbados or any Rhum Agricole from Martinique (additive sugar is illegal for rums produced with the AOC from Martinique).