Quoted:
Is it normal for there to be an odor close to the heater when on?
You'll occasionally notice an odor for a while after lighting the burner. Typically, it gets a lot less noticeable after the burner has been running for a while.
The only other time you might notice a strong odor is when a tank is almost empty. The "odorant" they use in propane has a tendency to settle to the bottom of the tank, and more of it gets drawn into the heater when the tank is nearly empty.
Also the instructions has a bullet in all caps that says "LP-GAS CYLINDERS MUST BE DISCONNECTED FROM HEATER WHEN NOT IN USE!", so this means every time I turn it off I'm supposed to remove the propane tanks?
If you're using 1 pound disposable propane cylinders, it's a good idea to remove the cylinder when the heater isn't in use. The connection between the cylinder and heater is at full pressure, and doesn't always make a perfect seal.
If you're using an extension hose to a large propane tank instead, you can turn off the valve on the tank and let the heater burn until all the propane in the hose is consumed. The next time you open the valve, be sure to do a "smell test" for leaks on the hose and fittings before lighting the burner.