Electrical – Ecobee3 not turning on heat, but fan is on

I would make the assumption that the furnace and fan wires are "bonded" together somewhere between your thermostat and your control board, therefore sending 24vAC to both relays and turning both on.

I'm assuming you don't have AC since you don't mention it, so in this circumstance you're probably looking at needing four wires -- ground (C), fan on, furnace on, and a positive feeder (R) (to power the thermostat and to send back to the relays).

To make the house more future-proof, I'd run more wires than needed in the event AC is ever added or the system changes configuration.

I have casual knowledge of HVAC, but in my experience of designing microcontroller-based thermostats, this is generally a good idea.

The terminal circled in red should be labeled HUM-H, which is the humidifier (hot) terminal. This terminal is energized to 120 V when the system is running in heating mode.

The terminal circled in blue is EAC-N, which is the electronic air cleaner neutral. This terminal should be tied to the other neutral terminals, as well as the system neutral.

If you move the humidifier transformer wire from HUM-H to EAC-H, the humidifier will run whenever the fan is on. However, this also includes when the system is in cooling mode, which is not recommend.

What you could do, is install an override switch, which will allow you to manually turn off the humidifier during the cooling season. So you'd run a wire from the EAC-H terminal, to one side of the switch. Then connect the humidifier transformer wire, to the other side of the switch. This way when the switch is closed, the humidifier will run with the fan. But when the switch is open, the humidifier will not run.

If you don't want a manual switch, you could mess around with a relay or other control circuitry.

Using a relay, it could be wire up like this.

That way the humidifier will run whenever the fan is running, except when the thermostat is calling for cool.