Does it stop if the TV is turned off? No, it does not. But your Roku does have a function that will stop playback if the remote isn't touched for four hours. It's labeled Bandwidth Saver and should be enabled by default. You can't adjust the length of time, so four hours is how long it could continue to stream after you turn off the TV.
That said, some online providers will ask if you want to continue after some length of continuous playback without user interaction. But I can't say for sure which ones might, and what their time limits might be.
Dan Roku Stick (3600), Ultra (4640), Ultra (4670), Premiere (3920), Insignia 720p Roku TV, Sharp 4K Roku TV, Nvidia Shield, Windows 10 Pro x64 running Serviio and Plex on a wired Gigabit network.