Is it true that some people always have red eyes due to dry eye sindrome, even if they're being treated for it?

Yes: This is indeed true, but sometimes there is a treatable or reversible cause of the dry eyes such as a medication or supplement causing dryness. An example of such would be frequent use of Sudafed or other stimulant medications for allergies. Allergies themselves can also cause red eyes which would generally be itchy, not dry. Exposure to irritants at home/at work could also cause this.