Prevent display turn off in a Mobile Browser
There is a Working Draft for a new Wake Lock API, which doesn't seem to be implemented anywhere.
For now, you can take a look at NoSleep.js:
Prevent display sleep and enable wake lock in all Android and iOS web browsers.
It requires user interaction though, meaning you have to attach it to a user input event handler (eg mouse/touch).
Since most mobile OSs require user interaction to stop the screen from going to sleep this is impossible from a website. You'll need to make a native app.