No. You do not have to attend, or even be admitted to, the school where you take the ATI TEAS® exam. Many students test at a nearby institution, a PSI testing center, or at home with TEAS at ATI, then send their transcript to the schools they are applying to.
But check first. Some programs only accept TEAS scores from exams taken through their own campus or testing center, and some require a specific exam type. Contact every school you plan to apply to before you register so you do not pay for an exam they will not accept.
When you do test elsewhere, you will need to send an official transcript to each school. TEAS at ATI and TEAS at PSI registrations include one transcript credit; exams taken through a school send the transcript to that school automatically. See What do I need to know before I register?