'Unconsciously' means 'without being aware'. If you do something without full knowledge of what you are doing, your action is an unconscious one, you do it unconsciously. Examples: 'an unconscious reaction' : a reaction that happened unconsciously 'Perhaps, unconsciously, I’ve done something to offend her.' However, the word that we need in your sentence is the adjective 'unconscious' which means 'in a state like sleep because of an injury or illness, and not able to use your senses'. We are not talking about feelings or thoughts here: They found him lying unconscious on the floor.

When someone is "lying unconscious on the floor" it is clear that they are lying on the floor and that they are unconscious. If someone is "lying unconsciously" on the floor, the 'ly' ending could indicate that unconsciously describes the manner in which they are lying on the floor rather than their state of being unconscious. In other words it could be taken as 'they are lying on the floor like they would be if they were unconscious'.