On a site with paid wifi, how do I log out to save minutes?

That depends upon whether the WiFi hotspot supports a logout function, some don't, for obvious reasons. Take a look at this app, if the hotspot supports logout, you can do it through the app:

http://www.easywifi.com/iphone-manual

All the sites I've seen like this have a web page via which you sign in and sign out. It's almost certainly not something you do via the iPhone itself, per se (other than via the web browser); just shutting off WiFi or quitting Safari on the iPhone probably wouldn't stop the time counter. You should talk to the company/site that provides the service and ask them about the procedure.

Regards.

There should in that situation be a logout on the web page as well. You'll need to ask the provider of the WiFi service, since the necessary procedure will almost always be different from site to site.