Since Roku devices don't actually turn off, I'm curious as to what might be going on. Let's get some more information about your setup and see where that takes us.
What Roku device do you have? The model number would be very helpful, if you can provide that.
Some people connect their Sticks to the USB port on a TV, which could actually turn the Roku off if power to the USB is cut when the TV is turned off. But it's usually better to use the included power supply to power a Stick. If you have a Stick, that is.
Let us know the model (number if possible; it's on the device, if it's easy to get to) and how it's powered.
Oh, and this is basic, but it's still good to verify. Ensure your TV is on the correct HDMI port for the Roku.
DBDukes Roku Community Streaming Expert Note: I am not a Roku employee.
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Could you tell us a bit more about what the problem is that you're running into? How are you powering the device? Is the device plugged into the TV USB port or is it plugged into the wall with the provided cables and adapter? In addition, have you tried using another HDMI cable and switching HDMI ports to see if you are experiencing the same issue? What troubleshooting steps have you taken so far to try to resolve the issue?
Please keep us posted and we will be able to assist you further.