DLsitePlay can do playback in Sleep Mode (background), but some devices or browsers are unable to do continuous playback in Sleep Mode.
* For Android
Try using a different browser (the latest version of Google Chrome, Firefox, etc.)
If the problem persists, try extending your device's time period for entering Sleep Mode to see if it is able to do continuous playback.
* For iOS
Try updating to the latest version of iOS, and view on Safari.