I've slowly battled through to the point where I have a working Grails app on Heroku. I'm going to bed now so I want to shut it down so I don't continue to use up my 750 free hours of my Dyno. It is probably obvious but I can't find how to shutdown my Heroku web server and restart it again in the morning.
To shut down your Heroku app scale the web processes to 0:
heroku ps:scale web=0
See the Usage & Billing dev centre article: "A web dyno that is idled continues to accrue usage". I believe maintenance mode will continue to accrue time; it simply serves a static page to visitors instead of serving your app.
Keep in mind each app receives 750 free billing hours per month. Depending on your intended usage, this may mean you don't need to shut down (i.e. if you only plan to run a single dyno for the whole month).