I'm a little confused by that last sentence, but if you mean the Roku is not connected to your TV via an HDMI cable, then that is certainly a problem for using a Roku. (Or any other video device.)
On the other hand, nothing about a Roku should stop a TV from turning on.
If in doubt, grab the TV remote and press the power button. If the power button on the TV remote works but the one on the Roku remote doesn't, then the Roku remote probably just got unprogrammed for the TV.
When I mean "on" I only mean "on". I don't mean that turning it on will cause a Roku to display. (For possible reasons such as: Roku not connected, not powered, asleep, TV on wrong input etc.)
