Social media has permeated throughout the personal, entertainment, and business lives of millions of people, and although it may not directly impact your dedicated server, you can leverage its power for promotion, communication, and even technical management.
Whether you are running a web hosting company or using your dedicated server for a single large site, promoting it through social media is a great way to inexpensively spread the word about your services. Forming a circle of Twitter friends, creating YouTube server tutorials, and maintaining a Facebook page, are all great ways to promote your server.
You will probably not use social media as a primary means of communicating with customers and users, but you should at least have it as an option for those users who prefer it. You may find that you are more easily able to communicated with certain customers.
If you are the type that has Twitter on all the time, you might benefit from a service that will send you a private notification in Twitter whenever you have a server issue. You can also track security updates for important server software and cloud services using various social media. Another good use of social media for technical support is to create a Wiki specifically for server documentation, frequently asked questions, and other relevant data.
Like any tool, social media will only work as well as you make it work. That takes some extra effort, but the benefits can be tremendous in the long run. It is not right for every dedicated server, but it might be right for yours.