Try Puma server.

Passenger wasn’t properly disconnecting the database, and I couldn’t
find documentation for this. The strange thing is that ohana-api is
using passenger as well, but isn’t running into this issue.
Update Ruby version to 2.1.2
Update appsignal to version 0.10.4
Update pg_search to version 0.7.8
Update newrelic_rpm to version 3.9.3.241
Update jquery-rails to 3.1.2
Update bootstrap-sass to version 3.2.0.2
Update puma to version 2.9.1
Try simple puma config file.

To see if it establishes connection with DB.
Try clustered mode.
Use Heroku-recommended settings for Puma.

https://devcenter.heroku.com/articles/deploying-rails-applications-with-
the-puma-web-server

https://devcenter.heroku.com/articles/concurrency-and-database-connectio
ns#threaded-servers
Remove port from Procfile.

It’s defined in the config file.
Switch back to Unicorn with Heroku settings.

https://devcenter.heroku.com/articles/rails-unicorn
https://devcenter.heroku.com/articles/concurrency-and-database-connectio
ns
Update appsignal to version 0.10.6
Update newrelic_rpm to version 3.9.4.245
Add unicorn-worker-killer to tame R14 errors.

For some reason, the app is exceeding the max memory available. To
restart workers who consume too much memory, we can use the
unicorn-worker-killer gem, as described in this article:

https://www.digitalocean.com/community/tutorials/how-to-optimize-unicorn
-workers-in-a-ruby-on-rails-app
Remove newrelic_rpm and appsignal gems.

To see if it makes a difference with respect to Heroku H13 errors.
Remove unused tld_length from settings.example.yml
Add Librato instrumentation to measure performance
Remove rack-timeout gem.

To see if it helps with Heroku errors, per the last comment here:
http://stackoverflow.com/questions/17763683/error-r12-exit-timeout-using
-herokus-recommended-unicorn-config