Going to sleep and closing your eyes seem inseparable for most of us humans. Fish sleep with their eyes open, though their sleep is not the same as our sleep as half their brain is awake most of the time. Trying to go to sleep with your eyes open seems fairly impossible, though there are people out there who can.
Generally speaking, people who sleep with their eyes open do so because of some injury and physical problem. This is called Lagophthalmos. Often it occurs due to damaged facial nerves or scar tissue around the eyes. These people can sleep with their eyes open, but only because they cannot close their eyes.
There is another group of people who sleep with their eyes open, but for very different reasons. Many sleep walkers actually have their eyes open, though they are deep asleep. These somnambulists, as they are called, will walk and talk and do virtually everything we do while we are awake when they are asleep. Their eyes are open but no one is home. When they wake, they cannot remember what has happened.
So, while you can sleep with your eyes open, it is far better to do it the traditional way, with your eyes firmly closed.