Since the valve drain is usually open, I am wondering if placing the cap on the drain has made my water heater dangerous.
To hold water in the tank your plastic drain valve should usually be closed. It does not matter if the water is held in the tank by a valve or a cap. A metal cap would be better than a plastic. You could also drain the tank and replace the gasket in the valve, but that is a lot of work for little benefit. Had not though of that. The drain valve is the one at the bottom of the tank. If this is a pressure relief valve, yes it would be very dangerous to cap it. The pressure relief valve is located near the top of the tank.
Easy method, change washer in valve Right method: Turn off main , attach hose to existing valve, route out side to a point as lows with respect to furnace height, open valve and drain water, remove plastic valve. Sweat on new brass valve with torch, turn main back on or just keep bucket under to collect drips and finally Pressure will not build so the thing is not going to explode or any thing - i.e. you are safe E-mail me if you don't fully understand Joe
You said it is the plastic drain valve. The pressure relief valve definitely is not made of plastic. A cap on the plastic drain valve at the bottom of the tank is not dangerous. If the bronze relief valve near or on the top of the tank has a drain pipe on it and that is the one you capped then, yes, you need to remove the cap and replace the valve as soon as possible.
If it is the drain valve and not the pressure relief valve, you should be fine. The drain valve is not usually open, just when the sediment is drained out of the heater or when it is drained.