So here's a nice article on the ingredient they are removing in question, which is brominated vegetable oil.
http://www.snopes.com/food/warnings/bvo.aspThe gist of the article is that consumed in large quantities, this particular ingredient can have adverse health effects, which could require dialysis to remove large amounts of bromine your blood (and large amounts of bromine in your blood is not a good thing). The quantities in soft drinks are generally low, but for people who drink a lot of this stuff, it could lead to acute toxicity requiring medical care. The ingredient is used to suspend citrus-derived liquid, which would otherwise float at the top. So it could affect flavor, but chances are good that the soft drink manufacturers already have or will find an alternative that'll do the same thing.
The removal of the ingredient is happening because of consumer concerns over the safety of it. Chemists, toxicologists and other scientists also want to reinvestigate the chemical with new technologies and methods that were not available in the 70s, which is the last time brominated vegetable oil was seriously investigated. It seems to me the levels in the soft drinks were low enough to not be of much concern (unless you binge on soft drinks) for acute toxicity, but evidence has come to light since the 70s that even in small doses, it may accumulate over time in human tissue, the effects of which (if any) are unknown. Either way, the companies are responding to consumer concerns over the substance by phasing it out of their products. It definitely seems that some of the concerns are overblown hype, that's why Snopes rates the health claims over brominated vegetable oil as a mixture - there's hype, but that doesn't mean there's not some truth to it.