Does anyone know how to remove the price from the email invoice template?
Rather than editing the .phtml file you should define a new email template using System -> Transactional Emails -> Add New Template.
Amongst other advantages this allows you to edit the template HTML directly in the admin system. As a starting point you can load the default "New Invoice" template and then alter the content as you wish. Remember to give the template a meaningful name.
Now, to enable this template for invoice emails go to System -> Configuration -> Sales Emails and set your new template as the Invoice Email Template.