So I've seen New Relic's guide here at https://newrelic.com/docs/php/new-relic-for-php
I ran heroku run /etc/init.d/newrelic-daemon restart and it shows:
Running `/etc/init.d/newrelic-daemon restart` attached to terminal... up, run.1
bash: /etc/init.d/newrelic-daemon: No such file or directory
So it looks like the proxy daemon was not installed by installing the New Relic add-on.
Any help is appreciated!
Contacted New Relic about this. Looks like it's currently not possible to run on Heroku.