I get notifications from apps like Snapchat or Facebook Messenger. Does turning off Refresh affect these notifications?
Background App Refresh allows apps to download information in the background under certain conditions.
After you switch to a different app, some apps run for a short period of time before they're set to a suspended state. Apps that are in a suspended state aren’t actively in use, open, or taking up system resources. With Background App Refresh, suspended apps can check for updates and new content.
Notifications are pushed from the server to your device at the time of an event (new email, message, whatever) and are unrelated to Background App Refresh.