I just want to remove the box lines around every section of the invoice layout, I mean those around biller, customer, resume and items sections, in order to get a text-only printout.
You should be able to remove all this lines just with editing the css file in : /templates/invoices/default/style.css
An other possibilitie is to copy the directory default and rename it to for example parralestemplate and modify this template to your needs. Now it should show up in the template selection list.