Finally got it fixed! I opened a ticket with Linode and they helped me get IPv6 configured. My situation is a little more complicated because I have two public IP’s on my server. It was the second IP that has the SSL certificate that had the renewal problems. For additional IPv6 addresses, Linode provides customers with a pool, not a single address and you have to pick an IP from the pool and configure it as a static IP. That was my first issue. Not being very familiar with IPv6, I didn’t know how to pick an IP from the pool and Linode explained that to me.
Now the fun starts. I’m running Ubuntu 16.04, so I have to edit the interfaces file, reboot and let Ubuntu create the resolver file with resolvconf. I’m using SSH to access my server, so I couldn’t bring down eth0 and bring it back up again. I could if I used Linode’s LISH console, but it’s clunky. Anyway after rebooting, Ubuntu wasn’t creating the resolver file - it was empty! I tried various things, but it seemed that as soon as I defined the second IPv6 IP in the interfaces file, Ubuntu couldn’t process it. I finally fixed the problem by disabling the resolvconf package. This allowed me to create a resolv.conf file manually. Voila!
When I tried to renew the certificate after setting up the interfaces file and resolv.conf, it still failed, but it was a new error message. I needed to fix one last thing: Nginx. I had to edit my Nginx server files to listen on the IPv6 addresses. As soon as I did that, the certificate renewal worked like a charm.