nginx-rtmp can take a source and forward to multiple outputs.
Fairly more advanced is BBC's Brave which is more of an API-driven live video editor that can push to multiple outputs.
It's essentially nginx-rtmp as a service with an SLA (maybe?)
an sla and a nicer management ui for handling keys and possibly reencoding for different services
Unfortunately it hasn't been updated for RTMPS (rtmp over tls), which is required/heavily encouraged for FaceBook Live.
You can work around this limitation using stunnel.
Just search for rtmps and stunnel. Quite a few helpful results. There is even a docker container that makes this pretty simple.