You can't delete ringtones that came pre-installed in your iPhone, but you can remove the third-party ringtones you've purchased or created. iTunes offers straightforward ways to delete ringtones either temporarily or permanently; either way, you can only do so on your computer using iTunes and not directly from your iPhone.
Launch iTunes on your computer and then connect your iPhone using the cable that came with it.
Click on your iPhone when it appears under the Devices listing on iTunes' left pane.
Select the "Summary" tab above the main pane on the right. Check the "Manually manage music and videos" option.
Click the little arrow to the left of your iPhone's icon. This reveals a number of folders underneath for the different media and books currently stored on your device.
Click on the "Ringtones" folder under your iPhone to display a list of its resident third-party ringtones on the right pane.
Click on the ringtone you want to delete from the list, or control-and-click to select multiple ringtones for removal.
Hit the "Delete" or "Backspace" key on your keyboard to delete your selected ringtones. Select "Remove" when prompted. This immediately deletes unwanted ringtones from your iPhone.
Select "Ringtones" under Library towards the top of iTunes' left pane if you want to permanently delete unwanted ringtones and prevent them from reappearing on your device after your next sync.
Click on the ringtone you want to delete from the list on the right pane.
Hit the "Delete" or "Backspace" key on your keyboard to delete the selected ringtones. Select "Remove" and then "Move to Recycle Bin" or "Move to Trash" on the next two pop-up windows.