Increased request latencies, increased heap footprint and increased CPU usage. I suspect it will probably be a bigger problem for people who use mobile.
Ideally, converting from http vs https should be transparent. There isn't any reason that I can think of to hard code a scheme in the urls on any site, except for explicitly switching between http and https for cases such as authentication.
Any other time, you can just use a relative url and the browser will pick up and use the preexisting scheme regardless of whether it is http or https. A single link which says "encrypt all traffic in https" and then redirects to
https://mises.org or
https://whateverpageiamon should make this possible.
So as long as you don't enforce either in any of your urls, you should be fine. Just put an encrypt all traffic button or something similar that redirects to the current page using https instead of http. Of course, if you have existing urls that specify specifically http, those will have to be removed to only specify a host and a document and not a scheme, but they shouldn't have been there in the first place. There's no point in writing
http://mises.org/daily when /daily works better.