Generally, App Inventor apps do not run in the background Herin.
When the device screen closes, the apps stop responding.
Specifically what do you mean by does not work?
From the documentation Texting
If the ReceivingEnabled property is set to 1 messages will not be received. If ReceivingEnabled is set to 2 messages will be received only when the application is running. Finally if ReceivingEnabled is set to 3, messages will be received when the application is running and when the application is not running they will be queued and a notification displayed to the user.
Is this the behavior you observe?
What value do you have ReceivingEnabled set to?
If the app is sleeping, nothing will happen unless you set this to 3 and you will only will get any messages after the app awakes again. You will not get a notification.