That really depends. We’ve designed Screenly Pro to cache all cachable assets. As such, if your internet connection goes down, all cachable assets (i.e. images and videos) will keep playing just fine. If you are using web pages in your playlist, these pages will be silently skipped while the internet connection is down.
It should however be said that Screenly Pro is only designed to survive shorter outages. If the outage lasts for several days, this will eventually affect the devices.