How to delete/remove custom ringtones (tones) from iPhone in iTunes 12.7?

If you tick the box "Manually manage music and videos", and click the "Apply" button, you'll be able to select a ringtone from your device (in iTunes) and hit your delete key (you'll be prompted for a confirmation to delete the file).

In iTunes on the computer.

1. Select the Device
2. Select Summary
3. Check the box next to Manually manage music and videos.
4. Select Apply

Now you should be able to delete ringtones using iTunes.

The answer is that it's impossible to remove them from your phone because any ringtones that existed on your phone after you updated to iTunes 12.7 appear to be inaccessible. They do not show up in iTunes in the "Tones" folder, so they can not be deleted. Even using the methods described above, turning on "manually manage music", restarting, syncing, etc.

Catalina and iOS 13

As of (and perhaps before, I'm not sure) macOS Catalina and iOS 13, there is no longer any iTunes app, so I know this answer does not exactly apply to the OP's question, but for those like me who found this question looking for an answer on how to delete custom ringtones from an iPhone under iOS 13:

On your iPhone:

• Open Settings
• Go to Sounds ➡️ Ringtone
• Scroll to see the ringtone you want to delete under "RINGTONES"
• Swipe left on the ringtone you want to delete, a "Delete" button appears on the right
• Press the delete button