In my active System Preferences / Network / Advanced / DNS, there are two servers listed first, ahead of a local machine and the OpenDNS servers. I would like to remove them, but I can't - they are grayed out.

I know what they are and where they came from, but why can't I delete them?

They're probably being auto-populated by your router. Try changing the DNS settings in your router/modem.

Nope. I rechecked that my router serves up DHCP but does not provide DNS servers. I don't recall why I set it up that why. Currently I configure each LAN node manually to use one LAN machine as DNS server. It runs BIND and forwards requests out to OpenDNS.

You probably need to clear your DNS cache if the router is just using the ISP supplied server and you are not using any custom settings in the Network pane.

Don't be afraid of the command line (/Applications/Utilities/Terminal)! Just flush your OS X dNS cache in Terminal with the command: dscacheutil -flushcache

Then restart.