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
Roku Community Streaming Expert
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