What is wrong with this script? [Unanswered]

You don't need to run a while loop for a function. A function is called upon what you said... in this case, when a player leaves.

Look at it like this, I join a game, finish my tycoon and decide to leave. Your code looks as follows:

game.Players.PlayerAdded:connect(function(player)
    -- code here
end)

So, I hit the X in the corner of the screen. All of your code inside of that first function is run, and I am now known as 'player'. So now you can see if me (player) has reset my tycoon and any other stuff.

Side Note

Instead of doing a seperate wait() inside of a while loop, you can write the code as while wait() do.

For all of those BrickColor.new("Medium stone grey")'s you can easily do BrickColor.Gray(). To check if the color will work for you (.Gray() does Medium stone grey), you can run this in the command line: print(BrickColor.Gray()). Use this Wiki page to learn more about BrickColors.

Anything else, let me know!

If it's a LocalScript, then PlayerAdded and PlayerRemoving doesn't work. Instead, use ChildAdded and ChildRemoved.

game.Players.ChildRemoved:connect(function(child)
    if child.ClassName == 'Player' then
        --Code goes here
    end
end)

Here's using the above in your script:

while true do
    wait(0.1)
    game.Players.ChildRemoved:connect(function(child)
        if child.ClassName == 'Player' then
            local player = child
            if player.Name==script.Parent.Owner.Value then 
                script.Parent.Owner.Value = "Nobody" 
                ------------------------------------
                local gate = script.Parent.Gate
                if gate.Transparency ~= 0 then
                    gate.Transparency = 0
                        gate.CanCollide = true
                    end
                ------------------------------------
                local co = script.Parent.OwnershipDoor.ClaimOwnership
                co.BrickColor = BrickColor.new("Medium stone grey")
                co.Parent.OwnerName.SurfaceGui.Owner.Text = "Claim Ownership"
                co.Defender.Disabled = true
                co.Claim.Disabled = false
                ------------------------------------
            end
        end
    end)
end