Extra headers like ‘List-Unsubscribe’ is not a function of the email server (exim4) but of the sending software. Ex: See php mail function here or PHPMailer library here
Furthermore I’ve found an old post about Exim… https://saeedrahhal.wordpress.com/tag/exim/ where the author says to setup the “headers_add” directive in Exim’s config file (but unfortunately Exim has several config files…)
I know this is a few years late , I’ve just been trying to work out the same thing without much luck online. I’ve found adding a filter at the end of /etc/exim4/system.filter works perfectly, see below:
# Add List-Unsubscribe header if not already present
if "${if def:h_List-Unsubscribe {no}{yes}}" is "yes"
then
headers add "List-Unsubscribe: <mailto:[email protected]>"
endif
Email went through fine and provides this in the header: