I'm not sure if this is correct, but I leave my baking stone in the oven at all times. I've never had a problem baking things besides pizza and bread on it (or with it on a different rack), and I usually never remember to take it out once it's cool anyway. I don't wash it very often but I am careful to cover it with foil if I think that something will drip down onto it. When I do wash it I don't use soap, but do use a little baking soda to scrub off stubborn residue. Good luck!

Why would you take it out? It adds mass to your oven, supposedly helping it to maintain a steady temperature, and you get to bake bread and pizzas on it. My own pizza stone has been in the oven for years. I scrape the carbonized crud off when food spills over on it, but otherwise I don't worry about it, though I do wonder about baking some things on the stone or on a rack.

The stone will absorb heat so the oven will take longer to heat up. If also has different thermal capacitance and conductivity, so any object placed upon it will react differently - basically the bottom of an object placed upon it will be hotter or colder (depending on the circumstances) than if it was just a wire rack.

Try the following (actually don't!)

1) heat oven to 200C.
2) place hand inside for 10 secs. OK?
3) place pizza stone inside to heat up (30 mins, say)
4) place hand on stone

Notice the difference? Your dish feels similar.

Just leave the stone in...not much will ruin it.

I also use one on my gas grill. Replicates a good pizza oven. I place the stone in the middle and crank up the grill with the lid down. It reaches greater temperatures than my oven (700 Fahrenheit +) like real pizza ovens that are even hotter.

Its porous so use just water to clean it. Or, like an iron pan or skillet, you can use course grain salt and oil as an abrasive to clean stuck on, and burnt food off of it. Because its porous you don't want to use soaps or detergents that may come out when cooking.