A point of contact from a company is being replaced by a new person, how can the new person take care of the old tickets.

You can change the email address of the requester from the Contact's profile. Add a secondary email address to the contact--> Make that the primary email address--> Remove the original email address and save the changes. 

By doing so, you have removed the old email address from the system and also given the new email address access to view the tickets raised the old point of contact.