There are several reasons why your Gmail isn't syncing with your iPhone. Go through the list and see what could cause this error and how to resolve it.
1. Your Gmail is facing a server issue.
2. Unstabled internet connection
3. IMAP settings of your Gmail have got disabled.
4. Incorrect server settings
5. Outdated Gmail App

See also - https://emailspedia.hpage.com/why-is-gmail-not-syncing-on-iphone.html