Some users that are running Android 6.0 Marshmallow and Nougat and up, have reported syncing issues related to Gmail and Inbox.
These tips could solve these problem:
Tip 1: Go to Setting --> Accounts & sync --> Auto sync, Make sure that you enable the Auto sync to make it work
Tip 2: For some people this works: Go to the global Sync setting: Can be found in Settings --> Data Usage --> Menu --> Auto Sync Data
Let me know if these tips helped for you.