I have heard that you can leave a pizza stone in the oven all the time, essentially storing it there. What are the considerations to keep in mind when doing this (type of oven, placement of stone, etc.)? I currently have a gas oven with the element located inside the broiler drawer below. Can I put the stone directly on the bottom of the oven, or should I keep it on the lowest rack? Are there reasons to remove the stone from the oven, if you're cooking certain things?
Actually I do this, and I do it because my oven is old and tempermental.
Adding a heavy heat-sink (like a pizza stone, or a half dozen fire bricks) to your oven will increase your pre-heat time, but it makes your ovens temperature much more stable. It's a good thing to do if you're planning on cooking anything that is really temperature sensitive.
If you're cooking something that is likely to spill over, you might want to remove it. They can be difficult to clean.
Otherwise, just make sure it isn't blocking airflow (possibly a problem with the very bottom of the oven, but depends on the design). I leave mine on the bottom rack all the time...
See also: What are other uses for a pizza stone?
Note that storing a pizza stone in your oven permanently will rack up your energy bill. As Satanicpuppy says, it's a heat sink that you need to heat up every time.
I do this as well. It's a good idea for ovens which have hotspots, or which are small/cheap and may drop too much in temperature when you open the door.
For a gas oven like yours, you want to put a rack in the lowest position and put the stone on the rack. Do not put the stone directly on the bottom metal of the stove; it will get thermal shock from rapid heating and crack. The baking stone will get a lot of carbon on it from food dripping and burning on its surface; turn it over about once every to months to minimize carbon accumulation.