The short answer is: nothing.
We have built Screenly Pro from the ground up to survive network outages, which is really the same situation as if our servers were to go down. This is an important feature, and it is particularly important if your nodes are using a 3G where network disconnections are common.
The only consequence of a node losing its connection to our servers, (for any reason), is that it cannot retrieve updates to assets and play lists. The screen will still keep showing the material that was last retrieved.
Beyond that, the nodes locally store the start and end time of playlists, (along with the asset, if it is cacheable). As a result, assets will still expire and stop showing as scheduled, as long as that expiration time was known at the time of the last sync.
Finally, it is very unlikely that our servers would go down for any extended period of time. We have many years of experience running web services, and we apply multiple measures to ensure high availability and uptime.