The impact of this depends on a variety of factors. We have designed Screenly Pro to cache all cacheable assets. As such, if your internet connection goes down, all cacheable assets, (i.e. images and videos), will keep playing. If you use web pages in your playlist, these pages will be silently skipped while the internet connection is down.
However, it should 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.