Restrict backups (for to iCloud or to a local computer) for the whole device.
Restrict backups for a specific app.
To see more info, take a look at the “Restrictions” section of the iOS profile creation dashboard. Also, you can select the backup options when you select an app to deploy.
We can enforce that the user cannot remove the restriction profile, but Apple enforces that the user can always remove the MDM profile (which will remove all the profiles that we added).
To gain complete control of the device (so the user cannot remove the profiles), you will need to use the Apple “Supervise mode”. See here