When I apply to add my website to a directory, it asks to add some script to my homepage or index page to validate the website. At what point can I remove this script? I want to add my site to various directories, but don't want to add lines of script to be there permanently. I'm assuming it is just for the initial validation and can be removed after submitting the site..?
That's going to depend from site-to-site. Some only want to initially validate your site to make sure it is legitimate at the time of submission. Others will require it as they periodically check to maker sure your site is still there as they try to keep their directory fresh and prevent dead links.
Considering how little traffic and SEO value these directories offer, I wouldn't be concerned with removing them. I'd rather have a tidy site then a mess just to keep them happy.