They suck. Its a package unit. If the green is a input to the board where is the output?
Do us a favor and take a picture of your thermostat connections, your unit's connections, and give us a link to a manual for the HVAC unit and the thermostat showing how they are supposed to be wired up.
Essentially your problem is that there is a low impedance path between one side of your transformer (presumably the red wire) and the other side of the transformer (connected to your green wire).
Now assuming that the green wire isn't hooked up to the other side of the transformer (if it is, then there's your short), proceed with the following instructions.
Do you have a voltmeter? With the green wire disconnected at both ends, and the unit shut off (insert safety blather here, just don't kill yourself), measure the resistance from each of the low voltage terminals to ground. If the resistance is low, say 0 - 50 ohms, then your control transformer is bonded to ground. This means that grounding out one of your thermostat wires (other than the white wire) will blow a fuse. Otherwise it won't.
It's a fifty-fifty chance that your unit does not have a grounded secondary (the 24 VAC side) and if it doesn't, then even if the green wire was "grounded" it wouldn't short out the transformer.
If it is grounded, and only if it is grounded, then keep the unit turned off, and measure the resistance of the green wire to the frame of your unit (which will be common to the transformer if it is shorted to ground) and measure the resistance. If it is in the tens of ohms (or lower), then it is indeed shorted. If you are lucky, you will have a spare wire that you can use instead of your green wire.
Now if your transformer isn't grounded to the frame of the HVAC, or if your resistance reading is well above 0 - 50 ohms (as in the megohms), then the "short" is either in the thermostat where the red wire and the green wires are connected, or the short is in the HVAC unit on the wire that the green wire hooks up to (presumably the fan).
The chances of it being shorted in the thermostat are slim, so I would start with the unit itself, so turn off the HVAC unit, connect the fan wire and the red together at the unit (just put a short jumper between the red and green terminals) leaving the green wire disconnected at both ends.
Now turn the unit on. If the short is in the fan relay, then you will blow yet another fuse, if not (and the HVAC fan runs when you turn the unit back on), then the problem is in the thermostat.