Just so you don't have to read this short article, the quick answer is No.
If you want to know why, read on!
The Reasoning
We recommend that you leave the pilot in your gas fireplace on at all times. The reasoning behind this is simple, we want to save you money and time!
The pilot on the fireplace is there for a few reasons. Firstly, it is an innovative method to ensure that you can simply press a button and have the luxury of heat.
The second reason is to prevent blockages in the pipes, ensuring proper gas flow. When the pilot is turned off, the smell of mercaptan (the chemical which is added to natural gas to give that notable odor) remains in the pipes. Little critters such as spiders are attracted to the odor of mercaptan and will crawl into the pipes of your fireplace and build a nest. This clogs the pipes preventing the flow of gas when you next turn on your fireplace. Unfortunately this will result in having to pay for a service call from one of our technicians to unclog and re-light your pilot.
So, when thinking of whether or not you should turn off the pilot you have two things to consider. Do you want to save the minuscule amount of energy that is burned and shut off the pilot, or keep it on and have a ready to go fireplace.
The choice is yours, but remember, if you are looking to avoid that service call next time you want to turn on your fireplace, then keep your pilot on.