Help getting started with Roku devices, including setup, connecting your device to your TV, linking a Roku device to your account at my.roku.com/link, adding channels, subscribing to services, and more.
No Roku player has an "off" function. As long as they have power, they're on. I think the one exception to this was the older Roku 4 (model 4400). Roku players use so little power, it make more sense to leave them powered on. One, they update themselves each day when you aren't using it. Two, electronic devices in general last longer when they remain in a constant power state. Shutting down and powering up applies a bit of stress to electronic components, so leaving them on helps moderate that.
If you are concerned with the player continuing to stream (and using bandwidth if your Internet has a data cap), simply make sure you press Home button on the Roku remote to stop any streaming and close any open channels. Then you can turn the TV off.
Dan Ultra (4640), Ultra (4670), Ultra (4800), Premiere (3920), Insignia 720p Roku TV, Sharp 4K Roku TV, Nvidia Shield, Windows 10 Pro x64 running Serviio and Plex on a wired Gigabit network.