I have to admit I've always labored under that assumption and so have other experts, like Dr. Jim Sears, one of the co-hosts of the popular daytime show The Doctors and Dr. Lisa Hark mentioned it my my article on eating foods to help prevent colds and flu.
Today I came across an article in USA Today questioning that belief. Some experts tell us that the evidence from 1973 isn't all that compelling (actually it was all done in the lab - not in the body) and that no research since that time has been able to back up the claim. A quick search or two in PubMed doesn't bring up anything, so maybe the idea that eating too much sugar can increase your risk of getting colds and flu is overblown.
Since I was searching through PubMed, I decided to look at obesity as a risk for influenza and found an interesting article from Europe that examines various risk factors for dying of H1N1 influenza. The authors found that obesity appears to increase the risk of dying from an H1N1 infection (though it doesn't say anything about obesity and the risk of catching H1N1).
The authors don't know why being obese could increase the chance of dying from H1N1. It could be due to obesity complicating the treatments used in severe cases, or it could be due to diabetes that occurs commonly alongside obesity, but there is also the possibility that obesity affects the immune system directly. According to a study in 2007, obesity messes with the immune function of mice. Research will have to be done with people to understand more about obesity's role in H1N1 mortality.
So I guess in a round about way, I might still believe sugar is bad for your immune system, but only if you eat so much of it you become obese. A little bit of sugar is fine, as long as you stay within your daily calorie need and get all the healthy foods you need too.