I have a brand new DIR-1960 which is also experiencing this same issue. I run my own DNS server on my internal network so I've entered its address as DNS #1. I then select or not select to relay DNS, both options fail and the log fills with the error messages reported above. The router has firmware version 1.11. So far as I can tell, the router has no built-in diagnostic capability; I can't ask it to ping or traceroute, for example. Unless someone knows how to get a shell...
The router's uplink port is connected to a cable modem and the link works. One of the LAN ports is connected to my local network switch which is in turn connected to the DNS server and other local machines. Externally generated pings work in all directions and all clients, both wireless and wired are able to reach the Internet via the cable modem. However, the DNS failure seems to cause the router to stop functioning after some time, last night about 0300 for example. Is OpenWRT the only way to make this thing work reliably?