I want to delete an iCloud account from my iPhone. It says I have to turn off 'Find my iPhone'. I can't do that because the applied on the iPhone is an old one to which I don't have access anymore. What do I do?

If the old ID is yours, and if your current ID was created by editing the details of this old ID (rather than being an entirely new ID), go to https://appleid.apple.com, click Manage my Apple ID and sign in with your current iCloud ID. Click edit next to the primary email account, change it back to your old email address and save the change. Then edit the name of the account to change it back to your old email address. You can now use your current password to turn off Find My iDevice, even though it prompts you for the password for your old account ID. Then save any photo stream photos that you wish to keep to your camera roll. When finished go to Settings>iCloud, tap Delete Account and choose Delete from My iDevice when prompted (your iCloud data will still be in iCloud). Next, go back to https://appleid.apple.com and change your primary email address and iCloud ID name back to the way it was. Now you can go to Settings>iCloud and sign in with your current iCloud ID and password.

Try contacting Apple for assistance resetting the password for the old ID by going to https://expresslane.apple.com, then click More Products and Services>Apple ID>Other Apple ID Topics>Lost or forgotten Apple ID password.

You're welcome.

Glad you got it fixed.