I have a question about the wiring. I have a Peerless oil burner and I am going to be using Ecobee4 thermostat to control it. I haven't installed the thermostat yet.
I am not sure what W/B wire is and what Y/O is. The instructions don't have these wires listed and the backplate doesn't have these letters on it. The existing thermostat works just fine with these wires but I am not sure the purpose of them.
You may find more information on the Ecobee web site. Ecobee4 section. There are helpful pages on things like connections and how to identify signals. https://www.ecobee.com/installing-your-ecobee4/
Download the full manual from the site. Then look at the Guide 2 section. You do not have a C connection mentioned in your post. The unit needs a C connection for power or a Power Extender Kit if your original thermostat did not use C or have a spare wire connected to C on the furnace end. They do make a Power Extender Kit (PEK) you maybe able to use. Depending on how many wires you have from the furnace to the thermostats location.
Do you have the part number for the thermostat you are replacing? That may give us information on what their terminals are used for.
Is their possibly a control board or connection box between the thermostat location and the downstairs. Since the Yellow RH does not seem to be used anywhere in the downstairs connection information you provided.
Maybe a system with more than one Zone and a Zone Controller?