If a website is running on HTTPS only, is it reasonable to close port 80 of the web server and only leave port 443 open? Does this action provide more security for the website?
Almost all sites leave port 80 open and have it issue a redirect to https://...
The reason for this is that is your site is
https://mysite.com/ then most users will access the site by typing
mysite.com and their browser will initially try
http://mysite.com/ If you've closed port 80 then their request will fail.
The security merits of this have been discussed in this question. It is more secure to disable HTTP, but for most sites the reduction is usability is not worth it. HTTP Strict Transport Security is an option to consider.