When on WiFi, you can manually set IP information, Settings > Wireless & networks > Wi-Fi Settings > Menu button > Advanced. Mind you, it would be very easy for anyone to modify those settings.
As far as when connected via cellular, I do not know of any way to modify those DNS settings as they come from the carrier and the carrier's towers. I'm pretty sure they don't want their users changing DNS settings which could help them bypass certain restrictions they normally have in place.
If it was feasible, you would need to do some rooting and hacking of files (e.g.
/system/etc/dhcpcd/dhcpcd-hooks/20-dns.conf). Supposedly there are DNS files that exist and can be modified, but I wouldn't recommend going this route.