Your link is to the old 1.x branch, here are deployment instructions for 2.2:
and play 2.1+ can serve https as well http://www.playframework.com/documentation/2.2.x/Configuring.... In practice, play instances are behind a front end server such as nginx for load balancing, https, caching, and virtual hosting. Play's convention for stateless sessions makes it really attractive to scale by adding more instances to the node balancer.
http://www.playframework.com/documentation/1.2/deployment