Personally, I try to not work under assumptions.
Obese people do live shorter lives. See:
http://news.bbc.co.uk/1/hi/health/511421.stm
The extra years are almost always cut off from retirement, when the individual would be taking out of social security and pension funds while putting nothing back in. Thus, obese people probably save society money.
Now if you want to find a group that costs society money, consider the childless. Because they won’t have children to support them in their old age, there is increased risk of their becoming a public charge. Plus they increase the ratio of retirees to workers, which must have the effect of either increasing taxes or reducing retirement benefits. So we should urge everyone to have kids, right? Despite being a thoroughgoing anti-Malthusian, I think not, but that is where some of this kind of thinking would lead.
P.S. The BBC article does not differentiate between the life expectancy of obese persons who are try hard but fail with diet and exercise, obese persons who resist all that, obese persons who permanently lose weight due to weight loss surgery, obese persons who undergo unsuccessful weight loss surgery, and the, I suspect, smallest group, obese persons who actually lose weight and permanently keep it off due to diet and exercise. Therefore, this kind of article provides zero evidence that we should “do something” about obesity.