No. In general, the guest who signed up on a device will get push notifications even if the app is closed or running in the background. If they click on the notification from the notifications list on their device, they'll be taken into the app.
A guest that logs in on a device that they did not sign up with will not get push notifications when the app is closed, only when they are logged into the app on that device.
NOTE: Functionality may differ depending on your app build and version. For details contact your Punchh Customer Success representative.