Yes, allergic rhinitis can and does progress and worsen into chronic sinusitis. Sinusitis hardly ever occurs without the sufferer first suffering from rhinitis. This is thought to happen when there is nasal sinus overflow obstruction. This leads to the colonization of bacteria and infection which then turns into chronic sinusitis.
There is an excellent possibility that allergic rhinitis can progress and worsen to chronic sinusitis, despite the fact that sinusitis is a bacterial infection. Sinusitis seems to always come when the sufferer is having inflammation from rhinitis. It could be that the inflammation and blockage that comes from allergic rhinitis make a good host for the bacteria of sinusitis when they invade.
I think that allergic rhinitis can progress and worse into chronic sinusitis. I think that if left untreated then allergic rhinitis can get worse and develop into a worse condition like chronic sinusitis. That is why it is important that those affected by it, make sure they do all they can to treat it.
Inflammation in the body can become systemic. Inflammation can beget more inflammation if it is not controlled by anti-inflammatory agents in the body. Allergic rhinitis can cause an inflammatory response that weakens the body's ability to fight inflammation due to the lowering supply of anti-inflammatory agents. Over time, the body will switch to a pro-inflammatory state which can lead to chronic inflammation.