How to delete an old Apple ID

Delete it from where? If you mean that you have a phone that is logged into someone else's Apple ID, the only way to remove it is by entering the correct login information. If the family member doesn't remember that, start here:

If you forgot your Apple ID password - Apple Support

You may also be able to have Apple remove it if you have proof of original purchase.

You might find this helpful How to delete your Apple ID account - Apple Support

I hope you find this helpful!

If you mean deleting an entire account and everything associated with it, carefully read this document --> How to delete your Apple ID account - Apple Support