If I move my SQLdata file from my F drive, delete the F drive, and rename the new Drive F will I have any file path problems?
If I move data from the F drive to a new drive, delete the F drive,
  and rename the new drive, will I loss the file path?


If you're going to move everything on F:\ to a new underlying disk and rename that disk F:\ then you should be okay, but you need to ensure all permissions are correctly migrated to avoid issues during restart.

The basic process would be:


Stop SQL Server
Copy all data from F:\ to new drive using robocopy or similar tool that can bring permissions as well. 
Remount F:\ as a new drive letter
Remount new disk as F:\
Restart SQL Server and verify health


Reverting should be as simple as reversing the drive letter changes if you run into issues.