Why isn't this GUI tweening properly?

It is actually your repeat statement. When you tween a Gui, it doesn't wait until the tween finishes to move on, the script moves right ahead. What is happening is that it is constantly calling the first tween position, resulting it ending up in a position that the tween is less than a pixel, thereby making it immobile.

Here is the solution I came up with:

---------------------------
function CreateLoadingBar()
local LoadingFrame = script.Parent.TextLabel.LoadingFrame
local Found = LoadingFrame:FindFirstChild("LoadingBar")
if not Found then
    local LoadingBar = Instance.new("Frame",LoadingFrame)
    LoadingBar.Name = "LoadingBar"
    LoadingBar.Size = UDim2.new(0,50,0,35)
    LoadingBar.Position = UDim2.new(0,340,0,0)
    return LoadingBar
    else return Found
    end
end
---------------------------
local Finish = UDim2.new(0,0,0,0)
local Start = UDim2.new(0,340,0,0)
Loaded = false
---------------------------
function Mover()
    wait()
    LoadingBar = CreateLoadingBar()
    --print("Loading")
        LoadingBar:TweenPosition(Finish,"InOut","Linear",0.5,true)
	repeat
		wait()
	until LoadingBar.Position == Finish
        print("Loading Bar hit Finish,putting back to start")
        LoadingBar:TweenPosition(Start,"Out","Linear",0.5,true)
	repeat
		wait()
     until LoadingBar.Position == Start
            print("Loading Bar hit start,putting back to finish")
        LoadingBar:TweenPosition(Finish,"InOut","Linear",0.5,true)
        
	if Loaded ~= true then
		Mover()
	end
end

spawn(Mover)
print("Done")

what happens here is that, at each transition, it waits until it gets to where it is needed, then moves on. If Loading is false, then it will repeat itself. Hopefully this was helpful.