So, we've been having trouble with our furnace for a couple of years now, and we've had a tech come out twice without getting anything fixed. Here's what it does: The unit tries to start when the thermostat tells it to, i.e. it goes, ahem, "vro-" and cuts, no ignition, no flame, no fan, no nothing. It does this maybe 30 % of the time immediately after a reset, then continues getting worse until it won't start at all. Flame sensor has been replaced and the board has been tested. We're also not getting any error code from the board itself.

One of the techs last year replaced the thermostat with a cheapo unit, which seemed to fix the issue for a bit... but it has returned. Is it possible the thermostat is just a piece of junk? One weird thing of note: one of the three wires that runs from the board to the t-stat isn't connected to anything at all, it's just... hanging there inside the furnace. I have the wire diagram and manual, but I still can't make sense of where it's supposed to go -- but I do think it's possibly supposed to start a fan. It's the green wire.

Any help would be appreciated. Like I said, a few hard resets and it'll get kicking again, which makes no sense really--but it never lasts for long.

OK first is the green wire a t-stat wire or just a green wire in the unit? Second does your unit vent with metal or plastic pipe?

turn off the power and disconnect the wire that is attached to the w terminal and the r terminal at the unit. This removes the thermostat from the equation.
Take a small wire and attach it from the r terminal to the w terminal. This will mimic a call for heat. Turn the power back on.
First the exhaust fan should start. Next the igniter will glow. Next the gas valve should open if none of the safety switches are tripped. Then the fan comes on. This is the circulating fan and it is independent of the green wire that seems to be left hanging.

@Joshdta: The unit vents with a metal pipe -- the green wire is attached to the board with the R and W t-stat wire, and then just dangles there.

@elephantears: jumping out the thermostat seems to mimic the response you describe--what exactly does that mean? Is the exhaust fan being tricky? Initially when the repair guy came by, that was his first reaction and it seemed to do the trick (although obviously this isn't a permanent solution) is it possible my t-stat is just installed funky?

The weird thing about jumping out the t-stat is that it never feels like I can guarantee if it's going to run through the false starts that it often does, does that make sense? I've jumped it out at the t-stat and had it NOT work, but doing it at the furnace seems to usually do the trick. Last year I had trouble and the house ended up dropping to 40 degrees before it started working again after a few dozen hard resets--obviously not a trick I'd like to repeat.

Actually, scratch out everything I said above -- I tried removing the t-stat from the equation again and I'm not getting anything at all. So now I'm thinking it's possible not the t-stat...

Dude, if you jump out the thermostat, (or mimic the call from the thermostat with a jumper wire) the unit should heat as long as the jumper wire is left between r and w. If it does, wire the thermostat back the way that it was because the problem is not in the thermostat or the wiring to the thermostat.
Now pay attention. If the problem is not in the thermostat, the next most likely place is the vent switch. This switch can also be jumped out but only for the purpose of trouble shooting.
The vent switch makes certain that the vent motor is blowing out enough exhaust air to adequately and start the burn. If the vent switch does not start open and then close when the fan motor ramps up to full speed, the switch opens to intercept the 24 volt signal to the gas valve.
Is the metal vent stack free from bird nests? We have found bird nests built in the pipe and even baby birds inside the vent motor assembly.
You can take it apart if you are careful to see how it goes back together.
You can find the wires to the vent switch and take them off. Start the unit and afterthe vent motor comes on, touch the wires together and see if it fires then. Remember this jumper must be removed after the test or unsafe operation may occur.
Changing the switch never helps. It is almost always correct in determining that the exhaust airflow is in adequate.

Climbed up on the roof to check the exhaust and it's completely closed off, so I don't see how anything could have got in there.The manual says I should get an error message of 3 blinks with a vent blockage, but I'm not getting anything.

Does the light on the control even come on at all when you turn the power on to the unit?

Just an update for everyone that helped (and I really appreciate your help). I had a guy come out today because it's starting to get too cold to deal with this, it turns out it was the circuit board, specifically the relay on the board, which we discovered by hitting the relay with a hammer and having the furnace kick on. Very bizarre, the tech said it was rare, but it happens. The good news is that after a slew of repairs over the last three years (according to our landlord anyway) I finally found a HVAC repairman in my town that is awesome (and did the estimate for free).

Good for you. I have started having boards like that repaired after I replace them and am able to save my customers money that way. I just replace the board and then have the bad one repaired and keep it as a spare for the next similar unit problem.
Those relays on the boards can be replace for a fraction of the cost of replacing them