Push notifications are visual indicators in a console that show when a record or field has changed during a user's session. If you use the Salesforce (Service Cloud) Console, you may want a visual indication when Case Flag is changed. Formula fields aren't supported "push notification fields", but you can monitor the Case Flag Date/Time field instead.
Push notifications are based on the Salesforce Streaming API, and that doesn't support formula fields. This is documented here:
As such, its not possible to get the Case Flag image to change (including appear/disappear), but you can have the record highlighted when the flag is set or cleared by monitoring the "Case Flag Date/Time" field which drives the formula fields in Case Flags.
For more information, see: