How do you clear the DNS Cache in your Asus 1900RT router?

Step 1:
Login to your router and enable SSH. You will find the SSH related settings at [http(s):// System (Tab).

You should see something similar to the screenshot below:

Click Apply at the bottom of the page after you enable SSH.

Step 2:
Login to you root through a terminal:

$ ssh -l admin
admin@'s password:

Step 3:
Issue the below command to reset the DNS Cache.

admin@(none):/tmp/home/root# killall -1 dnsmasq

Your router cache should be reset now. Alternatively, you can reboot the router to reset the cache.