Millivolt fireplace turns on by the thermostat but will not shutoff

Hard to say without seeing it, but typically you can use the existing flue provided you add the proper direct vent chimney insert for your particular gas fireplace. Alternatively, you can possibly knock a hole in the back of your chimney and just go out the side if you'd prefer (more labor, but less flue inserts you'd need to purchase).

UPDATE:

We actually did this recently. We ended up taking a direct vent unit and have the exhaust going through the existing flue (about 20'). For intake air, we're letting it take it from inside the house (since this isn't a primary heating unit, the loss in efficiency isn't a big concern for us). We've ran it a few times and seems to be working just fine.

Hopefully your issue may just be a faulty wall switch or a short in the switch wire. You can check this by jumping the TH and TH/TP terminals on the gas valve. If the unit starts, this is your issue.

Another cause may be a dirty or faulty thermopile (the largest of the 2 sticks on either side of the pilot hood. A thermopile is basically a series of thermocouples put together to create a higher voltage (which will control the main valve section of the gas valve. A proper thermopile should give you above 250 millivolts when the pilot is on(250 being close to the minimum that most gas valves will be able to operate at). First try cleaning the thermopile. Then remove the 2 leads off the gas valve and take a reading between the red and white wires of the thermopile. Anything under 250 millivolts and it must be changed. If the thermopile checks out then your issue is most likely the gas valve, and the gas valve will need to be replaced.