This issue is also occurring to me on my Galaxy S6 Edge in some circumstances:
- When only connected to my 4G network everything works as expected. (even with the screen off)
- When I'm connected to my Wi-Fi network at home or the Wi-Fi network at work, and the screen is off then the problem occurs and I receive no notifications.
Several other people are having this issue: (As this is my first post, I'm not able to post links yet, so you will have to prepend the http yourself

)
- forum.xda-developers.com/galaxy-s6-edge/help/push-notifications-stop-screen-off-t3071248
- forum.xda-developers.com/tmobile-galaxy-s6-edge/help/notifications-wifi-t3095156
People are reporting in these threads that the issue occurs only on IPv6 networks.
I looked into the option to configure a static IPv4 address in my WiFi settings, but that didn't solve the problem.
I don't want to root my phone to disable IPv6, so I couldn't verify that either.
My router doesn't support the option to disable IPv6, so I couldn't verify that either.