Will the site continue to work with the template if I remove the license key from the Wordpress platform?
Why would you want to?
If the theme author integrated license mean theme registration process in their theme then if you unregister the theme mean remove license then your site most probably will not run smoothly. You will not be able to auto update of the theme, theme bundle plugins and also will not get any support from the theme author for that website. Another possibility you will not be able to work with the theme demo import. Envato license mean purchase code activation is important thing to run any theme smoothly with all necessary resources.