I've just cleaned up that page a bit, in particular the OpenDNS and GoogleDNS solutions.
One of the points I added was that you need to be on as an administrator account, is that the case for you?
If you change it on your router, it will effect everything. If you change it on an individual computer, it will impact only that computer.
What DNS does is create an address book translating
www.name.com to an actual internet address like 123.122.111.1. In this case, Turbine's expired DNS resulted in it being pointed to an inactive status page. It takes a while for everyone to update their address books again.
So, what the OpenDNS and GoogleDNS solutions do is move you to a DNS that we know is already active away from your local ISP one that may not have been updated.
It should create no problems at all for you by leaving it like that. Many people use these DNS servers as their normal servers for a variety of reasons. However, if you wait a few days you could probably safely switch it back as everyone will be updated by then.