Help needed on thermostat wiring with Ecobee and Fast-Stat 5000
The Fast Stat device consists of two boxes, one located at your furnace and the other located at your thermostat location.  They connect to each other using your existing two-wire thermostat cable, and they provide 6-wire connections between furnace and stat.  You need to read the Fast-Stat documentation and make sure that the furnace and Ecobee are each connected to the Fast-Stat in a way that meets the needs of all three devices ... i.e. the furnace, the Fast-Stat and the Ecobee.
None of your photos shows the thermostat half of the Fast-Stat.  It might be buried inside the wall.  You need to find it, and you need to figure out which color wires are being used with it to correspond to the Ecobee's needs.
They have a web site with instruction manuals available for download.
If you have specific questions about how to apply the Fast-Stat installation instructions to your particular furnace, you can follow up here.  But I think it's probably already installed correctly ... you just have to find the other end and map out the colors.