I've been asking "Can you run a business without servers?" on a few LinkedIn groups and so far the response is largely doubtful with the most common response being "What if the internet goes down and you depend on cloud services instead of local servers?"
This is interesting because in my view having a poor internet connection provides an even stronger incentive to stop using servers. For example, if your local internet connection fails and you are running a local mail server, communication stops (except internally, if you are lucky). Compare this to when mail is managed by a cloud provider; you can pick up where you left off using a 3G phone, home connection or wifi cafe. Services do not depend on your flaky ISP being available and you are much less vulnerable to disruption.
That's mail servers, but what about print, login, and storage servers? I think we'll see authentication move from the operating system to the browser, particularly as our technology becomes more 'device' driven. That removes the need for login servers. Google's cloud print service is another example of a service providing what used to be managed by servers, and there's plenty of shared storage solutions available too.
So what will run on the work network of the future? My guess it will be just a bunch of devices. Network switches, firewalls, printers and various devices such as tablets with little more to them than a web browser. Where does that leave the IT department? Hopefully doing what they should always have been doing. Integrating services, supporting users, automating process and connecting the business with customers through online services. Oh, and swapping out those devices from time to time. What they won't be doing (and what they really should never have spent so much time doing) is spending their time patching and fixing server hardware.