A nice elaborate information is presented by sinusprobleminfo, for those who are not convinced with plain vanilla sentences.
Cold drinks are the worst of all foods. Cold or iced drinks slow the cilia in your sinuses. Why is this bad?
In a nutshell, mucus captures all the foreign substances to protect your lungs. and the cilia is the microscopic hair that sweeps out this into your stomach and nose. If this flow is stopped, mucus increases.
Your sinuses have a mucus lining. This mucus lining keeps the membranes lubricated, soft and pliable. The mucus lining captures foreign substances like, dust, pollen and even bacteria. It keeps it from getting into your lungs. This mucus then needs to be swept out of the sinus, into the nose and then the stomach where it is destroyed. The cilia, which are microscopic hairs lining your sinus and nose, sweep the mucus away with rapid back and forth movements.
When something slows them down, the sweeping of the mucus slows or stops. When that happens things start to get blocked and congested and you have the beginnings of an infection. Cold drinks, by slowing the cilia, can create conditions where you could more easily get a sinus infection.
If you already have a sinus infection, cold drinks will just make matters worse by slowing the cilia in a situation where they are already impaired.
So avoid Cold drinks if you have a risk of developing sinus! :)
On that note, there is another insightful article on the difference between Cold and Sinusitis - interesting read
Source: Article, Picture