Hmm .... unfortunatly, I don't know of any that are open.
In the late 90s, there was one in Mt Prospect Il that formed around 1997. They went out of buisness later the same year. They had about a dozen midrange PCs (at the time) all with 27inch TV's connected to them via Svideo. At the time I had only a 486. There were a couple of times where my friends and I rented out the whole place for a few hour blocks at a time and played Duke3D. It was a bit pricy, but holy hell was it a lot of fun..
There was an internet cafe/gaming lan place in Arlington Heights that closed just a year or two ago. My brother used to go to that one, but I had a decent system of my own and plenty of games and could go to private lans instead.