The DNS just tells places where you are so I don't see any real security issues.
It's precisely because DNS tells you where places on the internet are, that it does have huge security implications.
For example, a crime gang could setup (or compromise a legitimate) DNS server, and program it to send some requests for banking sites to their own fraud/Pfishing (fake) banking site. Since this redirect happens at the actual URL translation level (controlled by the DNS process), even entering the URL directly into your web browser wouldn't protect you. And criminal gangs have done this sort of attack in the past, and this is just one fraud/security attack that can be facilitated by corrupting DNS results, which makes choosing your DNS server very important from a security standpoint! i.e. You don't want "the bad guys" controlling where you actually visit if/when you try to visit some "high security" site.
Now, that said, most of us are fairly confident that both google DNS and OpenDNS are run by people that understand these DNS security issues, and take active steps to protect your internet access when you use their DNS servers.
And since googleDNS/OpenDNS also happen to be the DNS servers we usually recommend people try (to help resolve streaming issues on the ROKU), you can be reasonably confident that trying those specific DNS servers is not likely to hurt your internet security. i.e. While all DNS (including the "default" DNS your ISP runs) have potentially very serious security issues, the specific DNS servers run by google and OpenDNS are probably in the higher security category to begin with. In fact, I personally use google DNS myself, in part because I trust google's DNS security more than I trust the DNS security of my own ISP!
FWIW: When I last tested this, I discovered that my ROKU streaming was about as good with googleDNS as with my ISP. That being the case, I currently am running with google, as I trust the security of google better than I trust my ISP to prevent hacking of its DNS servers. Of course, DNS is clearly a YMMV thing (depending upon location, ISP, etc.), so while googleDNS works well for me, it might not work as well for others.