TaskRatchet uses the timezone you select to determine when your tasks are due. If you specify that a task is due at 6pm, the task is due at 6pm in your specified timezone. If you change your timezone, your existing task will now be due at 6pm in your new timezone.
You should see a “Changes saved” message. Once you do, your timezone has been changed, and all due dates will now be treated in reference to your new timezone.