Has Let's Encrypt been put into production with CiviCRM?
Yes - I'm using it on one production site and one pre-production.  One has the full letsencrypt client, but it assumes root access.  The other uses an alternative client https://github.com/diafygi/letsencrypt-nosudo that doesn't need root and works on shared hosting (but you may need to tweak your .htaccess file for the verification process).

I've not needed to renew certs yet.  Although not really a problem, be aware that certs only last 3 months (though I think that's during beta).  If you have access to the webserver config, there is an auto-renewal feature but if you need to raise a support ticket on a shared hosting platform, that's less convenient.