Windows Phone 7.5 and 8.0 haven't been supported in some time, so it should come as no surprise that Microsoft is shutting down push notifications for the platforms. The news comes from Microsoft's Answers forum (via Windows Central). This means three things: you'll no longer receive push notifications, Live Tiles will no longer be updated, and your device will no longer show up on "find my phone".
Microsoft says that notifications will continue to function for Windows Phone 8.1, and "newer phones running Windows 10 Mobile are still fully supported." It's worth noting, of course, that Windows Phone 8.1 support ended back in July, so there's no guarantee that Microsoft will continue to support it with notification services for any given time.
The original version of Windows 10 Mobile, 1511, has also been unsupported since January. If you're on this version of the OS, it's recommended that you upgrade to a later version of Windows 10 Mobile.
If your phone is stuck on Windows Phone 8 and you have one of the few devices that was never offered an upgrade to 8.1, there used to be a workaround called Preview for Developers to get the update. This has since been shut down as well.
The changes to the notification system will go into effect on Tuesday, February 20.