Snapshot on Always On Availability Group in suspect mode
Operating system error 665(The requested operation could not be completed due to a file system limitation)


This is a common issue with large sparse files as it causes NTFS metadata fragmentation. You're not hitting a SQL Server issue, you're hitting (as the error states) a filesystem limitation (NTFS).


  What can we do to avoid errors like this?


Not use snapshots. Not use NTFS (ReFS instead).


  Will changing to ReFS help or perhaps are there any other NTFS settings that can be ammended?


Yes, changing to ReFS should alleviate the issue, assuming you actually need the snapshots. Changing less data in the source database would also help alleviate the issue. 

No, there aren't really any other NTFS settings that can be changed.