Most fire pits have specialized safety mechanisms that will shut off during overheating. Although using a fire pit with a propane tank inside is relatively safe, handling open flames involves some safety risks.
Another good idea to check regularly - is the spark ignitor on your fire pit table. This safety mechanism allows a controlled "Spark" when the button is pushed to ignite the gas burner.
If the safety ignitor button is not functioning, you can bypass it by lighting the burner with a lighter. Still, replacing the push-button ignitor (also called a Piezo ignitor) is far safer and more accessible.
If you are particularly concerned about safety, seeking out units equipped with more robust safety mechanisms may be worthwhile.  Here is a good article from PG&E.com on the best Fire Pit Safety Tips when inspecting and using Gas Appliances (Pacific Gas & Electric).