How do I stop receiving notices that I need more iCloud storage?

Either subscribe for more or delete content. It’s just like the nag on your Mac or Windows computer if your hard drive is near full or full. Once the available storage is full, things don’t work (e.g. iCloud email) and data starts getting corrupted (think of your notes, contacts, calendars, or other data sync’d to iCloud suddenly getting corrupted), hence the nags. You either need to add more storage or clear out space to make the nags stop.

Buy more iCloud storage is the sure fire way to stop these messages. You can get a plan for $.99/month = $12/year and that gets you 50GB. The reality is if you want to backup to iCloud and you're running out of space in the cloud, you'll get these messages.