So while I was attempting to figure out how to get our fan to run independent of our heating/cooling, I looked at our thermostat wiring and discovered that it only had wires running to C, W, Y, and RC/RH. I'd read that it was common in older homes for the "G" fan relay wire to get used as the Common wire, so I tried swapping the wire from C to G to see if it would work (and popping in batteries onto the thermostat, knowing that it was no longer getting power from the C line).
But it turns out I missed the critical part where you have to re-wire the control HVAC side as well. So effectively, I likely ended up wiring the "Common" line to the "Fan Relay" line on my thermostat. I switched the wire back to the original position on the thermostat, but it looks like I may have already done damage, as the thermostat now no longer powers up without batteries, and with batteries, I'm unable to start up the cooling or the heating system.
Is it possible that I fried the control board on the furnace, just by attempting this wire switch?
Additional context: It looks like the thermostat is functioning properly, in that I can hear it "clicking" on to attempt to turn on the heating or the cooling system… but it just doesn't seem to trigger anything on the HVAC side.
Best Answer
How thermostat signaling works (and how to test manually)
First, a little background about the thermostat control: It's dumb. Really dumb. The signaling is done by simply shorting wires together. The R wire is shorted to W to call for heat, or R shorted to Y to call for cooling, or R shorted to G to call for fan. Stuff, like digital thermostats, can be powered between the R and C wires.
You can test the furnace controls without the thermostat. Simply short the R and W (or R and Y) wires together and verify that heating (or cooling) begins.
This can be tested right at the furnace control board too, avoiding all doubt about the thermostat wiring. Note that the control board is often found in the blower cabinet and you're going to have to defeat a safety switch to perform testing there. It's your responsibility to ensure this is done safely -- you don't want clothing, long hair, or other things being sucked into the blower while working here. There also may be high voltage terminals exposed, so be careful about what you touch. Bypass the safety switch with one hand and, with one or two other hands, touch a jumper wire (a paperclip will do) between the R/W or R/Y terminals on the control.
What probably went wrong
To call for fan the thermostat will short R to G. But you had C on the G terminal, so the thermostat shorted R to C. That's a dead short across the 24v transformer in the furnace. Either the switch in the thermostat will have failed, or something in the furnace will have failed. Control boards often have a small fuse, about 2 amps or so, to protect the 24v side of things. It could be that the fuse has been blown.
How to diagnose further
If the manual tests shorting R/W and R/Y didn't cause the furnace to operate then get a multimeter and set it for AC volts. Turn on all power to the furnace and safely bypass the blower door safety switch if appropriate. Confirm (measure) that full AC mains voltage is supplied to the controller. If it is, then measure voltage between R and C terminals. You should find something in the neighborhood of 24-32 volts. If not... we'll have to dig deeper by examining the control board for a fuse, testing continuity of the fuse, testing the output voltage of the transformer upstream of the fuse, etc.