I'm guessing the Roku is sending a signal in a resolution which the tv does not support. If that's the case, you should see the normal "bouncing letters" Roku startup screen, but lose your picture when it tries to load the home screen.
I'd suggest trying the Roku on a different TV if one is available, especially if it has a working Roku. If it comes up okay there, go into Settings > Display type and change the Roku to the actual resolution of the problem set instead of the "auto" setting. When you get to the countdown screen click on the "yes" option. Now move the Roku back to the first set and see if it works.
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