How can I log out of a locked Apple ID?

You can only sign out of an activation lock if you know the Apple id and password that was used to set it up. Without that information, you will never get past it. The best you can do is to return it for a refund if possible or recycle it at an Apple Store and then buy a phone that you can use.

Did you buy a used iPhone from someone?

If yes, you will need the Apple ID and Password of the person who originally set up the phone.....or.....have them log on remove the lock on the phone.

For the future......never buy a phone unless it powers on to the "hello" screen.