It's best to close the valve on the propane tank, and I'm sure the instructions tell you to do that just like any other fire pit or BBQ grill that I have seen. The valves on the fire pit are outside and exposed to the weather, and they are more likely to leak (in my opinion).
That said, many people do leave them on all the time, accidentally or on purpose. To answer the other part of your question, a propane leak from a tank like that outside doesn't pose a lot of risk. The gas dissipates into the air so it's not going to suffocate or poison anyone. Also, propane doesn't burn unless it's mixed with the right amount of air. Too little or too much and it won't ignite. Theres a small chance that just the right amount of propane could build up in the box where the tank is, and that could cause a small explosion that would damage the fire pit.
Now, all that is to simply say that it's not nearly as dangerous as a leak in your house, but there is risk. At the very least, you just paid $20 for a propane tank that you'll use once because of a small leak. Turn it off when it won't be used for a while.
You could if you have access to them to make yourself feel better. We have a stove with 3 pilot lights (it is a seriously awesome stove - don't judge!) and turn off one of the pilot lights because we never use that feature. Have had no issues although I do occasionally stick my nose there to make sure I don't smell gas.
And now you've just reminded me to turn off the pilot light on our furnace. Thanks!