If you simply turn your TV off while the Roku is streaming something, yes it continues streaming until the 4 hour alert is activated to ask if you're still watching. If you first press the Home button on your remote and see the Roku home screen before you turn off the TV, then in general the Roku is no longer using data. However, as long as the Roku is powered on and connected, it does occasionally check the Roku server for updates. Other than that, there's very little network activity from a Roku device.
You cannot disable the auto-update function, so if you want it to use absolutely no data when not in use, you need to power it off.
You didn't mention what model Roku player you are using. If you are using a player that has a USB cable for power, and you use a USB port on the TV to supply the power, then it's likely your Roku is being turned off every time the TV is powered off. If you see the bouncing Roku logo when you power the TV on, then it's turning off with the TV.
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I have 3 Roku Streaming Stick +, and all 3 are plugged into the USB port on the TV for power. 2 of the 3 when I turn off the TV, the Stick also turns off, but on one of them plugged into a Vizio TV, the Stick stays on most of the time. I did discover that you need to exit the streaming app such as Netflix when done viewing or else it continues streaming, and the next time you turn on the streaming app, it is now 4 hours in the future. This is bad. If the TV is powered off, the streaming should stop as well. I do not have unlimited bandwidth as Comcast changed it recently this year. What can be done to stop this, other than getting out of bed at night, and disconnecting the USB port.