Is Pokemon Go supposed to keep your screen on?

It doesn't, so you'll have to update your system settings.

On Android, the setting should be located somewhere like this, but might vary slightly depending on which phone you're using:

Settings -> My Device -> Display -> Screen Timeout

If your phone is like mine, you might find that the highest the inactivity timeout can be raised is to 10 minutes. If that's a problem, you can install an app to keep your phone from timing out instead.

Full disclosure: I am the author of the app I'm about to recommend. The link is below.

Wakey is an Android app that is built to keep your screen on, on your terms. I've worked very hard to make it work well and be simple to use, and gives you lots of options which may improve battery life while using Pokemon Go. It works better than the "Battery Saver" mode included in Pokemon Go, which only really works when the screen is upside down. Wakey works in any orientation.

You can choose a mode that will allow the best combination of screen and battery - you can choose CPU only (this allows the screen to shut off, but the phone stays awake. I'm not 100% sure this is compatible with Pokemon Go, but if not...), or you can choose "Minimal Brightness" mode, which lowers your screen to the lowest brightness, but keeps it on. Finally, there's "Allow Dimming" mode, which lets the screen go partially dim, but not fully dim or off. I would recommend trying the modes in that order, as they are listed in the order of most battery saving.

With Wakey, you can set it to keep your screen on manually, or as long as Pokemon Go is on screen (or any app or apps you choose), or as long as the device is charging (the Auto features require in-app upgrade to Premium, but everything else I'm talking about here is free).

Android users can download it here