How to erase data from an inaccessible phone that won't reset?

Can you erase data from a phone that cannot be accessed? It won’t get beyond asking to reset the factory settings and then it won’t reset. We’re getting a new phone, but I’m afraid to turn over the old one in case somebody can access the data that’s inside.

[Re-Titled by Moderator]

Original Title: Can data be erased from phone that will not reset to factory settings and will not allow access?