Apparently there's no way to actually turn the Stream Deck off as the 'sleep' function just dims the buttons to minimum brightness. Since it's an OLED screen you may expect burn-in to be an issue. But here's how you can prevent or at least minimise burn-in:

• In the Stream Deck programme go to Settings => General => Sleep After and set the shortest time out that you can live with.
• Use the lowest brightness setting you can (minimum is still reasonably bright).
• Select 'Add Screensaver' and then upload a completely black image (480 x 272 px). You can use the one linked below.

When the screen goes to sleep the black screensaver will be "displayed", and the keys will go (almost) blank, protecting them from burn in.

You're welcome.