Anyone with an app affected by this because you use Heroku's SSL endpoint can fix this by switching to SNI. Unfortunately Heroku's API is also affected. There is a workaround... just edit your hosts file to point to the proper ip addresses temporarily.
You would set your name servers with your registrar, such as godaddy. If dnsimple is your registrar as well as your DNS provider you may be out of luck until they come back online. The help page is here though: http://support.dnsimple.com/articles/setting-name-servers/
Free solution that worked for me: Set up a free account on cloudflare.com, duplicate all dns records (thankfully I have a simple setup)... but next time I will keep a backup zone file!
FYI - Instead of an Alias record on DNSimple, CloudFlare will allow a CNAME record for the root domain using "CNAME flattening".
You can now set CloudFlare's DNS service to "bypass Cloudfare" on all records by clicking the icon so you don't get any of their magic (unless you want it).
Then add CloudFlare's 2 nameserves to your domain as your first 2 name servers. No need to remove dnsimple's name servers.
Now you have 2 DNS providers in case one fails, just make sure the records are the same across them both!
add to /etc/hosts
23.21.149.112 api.heroku.com 107.21.99.123 ssl-doctor.heroku.com
Then use Heroku CLI to switch your app to the new SNI endpoint
heroku certs:add --type sni server.crt server.key -a YOUR_APP
That will spit out a new host name that you can point your DNS to and still be online.