I'm new to my Roku 4661R, 9.1.0 b4111-46, using Ethernet via a NAT router.
In scouring teh webbunetz:
1) Roku uses the router's DNS per the DHCP lease.
2) Roku OS uses Google DNS, overriding the router.
3) Roku OS uses the router's DNS and Google DNS as fallback.
4) Roku apps are responsible individually for DNS, overriding the router.
5) Roku apps can do DNS as a developer's option, overriding the OS and the router.
6) Something else I didn't spot because my eyes crossed.
Most of what search returns from in this forum (I stopped reading on page 6 of 320) and the internet (reddit is loaded; I'm not alone) is popular conclusion and empirical speculation, evidenced by the ubiquitous "I think..." lead in.
Seemingly nothing approaching hard documentation, from Roku or otherwise, but for "The Roku...relies on DHCP" in devrdander's May, 2008 posting here.
How does Roku do DNS?