Fireplace shuts off on it's own after approx. 20 minutes unless I keep it on a very low flame setting.

Sounds like a limit switch. What brand fireplace?

Vermont Castings - approx. 14 years old

Not familiar with the brand. Do you have a wiring schematic? Find out if you have a high limit(protects the unit from overheating) If the limit has become faulty, then if will open too soon, causing the burners to shut down until the unit has cooled down. There are other possible causes of your problem(many). I would suggest having a tech out to fix it.

Is it controlled by a wall thermostat, remote thermostat, or local thermostat(on unit).

No thermostat, I installed a valve in the line to control the flame level and heat output, and that has worked perfectly for the first 12 years. This issue has only surfaced in the last year or so. I have since moved the stove to my lower level and need to depend more on it's output.

Is the gas controlled by a millivolt system?

Possibly the thermopile that generates the necessary voltage, which sits in the flame is not being heated sufficiently by the throttled back gas flow. Or possibly the thermopile has just seen better days. I have one at home that is dead too just haven't need the fire place yet this season.

Another thought. If you do have a remote swich or thermostat the contacts on this might have built up a resistive layer over the years. After all with only a few hundred mV it doesn't take much resistance to have the gas valve drop out.

Did you find a high limit swich?