Roku's have a 'bandwidth saver' setting that stops an app after about 4 hours if you don't answer a prompt. If you don't worry about bandwidth, you can disable the setting. This also assumes a separate Roku device and not a Roku TV. I assume turning off a Roku TV will also shut down any running apps.
Thanks for the quick reply! Yes, this will be a Roku stick using the Comcast streaming app. That sounds good - if I understand you correctly, if I disable that app shutdown feature, the Roku will stay "hot and live" when I power off the TV, so it should come right back up to the same screen when the TV is turned on again. Is that setting easy to find?