The two potential threats caused by too much soda intake, are diabetes and caffeine overdose. A third could be gingivitis, but looking after your teeth with a brush at least twice a day will mitigate that threat.
Standard Brands: Now 50/50
These drinks usually contain sucrose as the natural sweetener. However, laws have since been in place that requires these drinks to cut down on sugar levels. How do they get around that? They cut down on sucrose and attempt to make up for it with an artificial sweetener such as aspartame. While sucrose levels are dropped by 50%, the overwhelming sweetness of the remaining 50% should be such that most people won't be able to taste the difference with the reduction.
Diet and Zero Brands: 100% Aspartame
These brands completely replace natural sugars with artificial sweeteners; typically, this is 100% aspartame, but can sometimes include sucralose. Aspartame is also used in packaged artificial sweeteners such as Canderel.
There have been varying degrees of study on the effects of aspartame on the human body, and most, if not all, conclude that it is safe for consumption. I believe the most commonly reported side effects are varying severities of headache; a symptom which also exists with caffeine overdosage and caffeine withdrawal.
Caffeine
Whether you go for the standard or diet alternative, Coca-Cola and its derivatives/competitors all contain caffeine. Typically, this is around 35 mg/330 ml. Reduced sugar alternatives actually contain slightly more caffeine. Tea, coffee (barring decaffeinated, of course) and energy drinks such as Red Bull contain over twice as much caffeine in the same volume, versus these soft drinks.
The approximate safe daily intake of caffeine is estimated to be within the 300–500 mg range, though medical conditions affecting your liver will decrease this safe zone. For a proper figure, you can calculate it from your body weight and a general idea of your metabolism. A generalization would be that 10 cans per day are within most people's safe limits.
Lethal doses of caffeine are estimated to be approximately 150 mg per kg (or 2.2 lb) of body mass. For an individual weighing 75 kg (165 lb), that's 11,250 mg, or roughly 321 cans of Coca-Cola. For an individual weighing 50 kg (110 lb), that's 7,500 mg, or 214 cans.
Just remember that you must take into account other sources of caffeine when calculating the safe daily intake figures, such as chocolate, kola nuts, medication, caffeine pills, tea, coffee and energy drinks.
A caffeine overdose can have highly negative side affects, including the possibility of death. Other side affects are fever, heart palpitations, rapid breathing, tinnitus, twitching, frequent urination (caffeine is excreted from the body via urine), hallucinations, seizures, headache, anxiety, insomnia (unsurprisingly) and hematemesis.
(UK) Sugar Taxes
Introduced in April of this year by the UK government, all soda vendors are now taxed on the following two bands;
> 5 g/100 ml @ £0.18/liter
>= 8 g/100 ml @ £0.24/liter
Naturally, this taxation on these companies has found its way to the consumer as well. Two examples, based on my experiences, are McDonald's has increased its sodas by £0.18, while the Co-operative has increased a 2-liter bottle of Pepsi or Dr. Pepper by anywhere from £0.40 to £0.55 (now selling for £2.50 per bottle).
Under the figures defined by the government, a 2-liter bottle of ordinary Dr Pepper, with 4.9 g/100 ml, should be free from this taxation, but you definitely notice the difference as a consumer...
TL;DR
It's no worse than tea or coffee, providing you look after your teeth and drink sensibly. An overdose on caffeine is the largest cause for concern, and this can be more easily accomplished with drinks that have higher caffeine content, so I think this warning should apply more so for those than sodas. Sodas without caffeine don't carry this risk.