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Once you change the IP address on the DNS to have the same string as the new routers string, then that would be thats all thats needed. You would need to change the Gateway address on the DNS as well to what the new address of the new router is. So if your old router was using 192.168.0.1 and the new router uses 192.16.1.1, then you would need to change the DNS IP address to 192.168.1.### and gateway to 192.168.1.1.
IF your using mobile apps that have configured addresses, these would need to be changed as well. Mydlink.com is not effected unless the main host router prevents communication thru it.