Whenever I change the value, it never changes to the other part of the script!?

You probably intended to have the loop outside the conditional, so that it's checked periodically, no? Like this:

FirstImage = "http://www.roblox.com/asset?id=2967292661" 
SecondImage = "http://www.roblox.com/asset?id=2967293159" 
ThirdImage = "http://www.roblox.com/asset/?id=2972903107"

while true do
    if game.ReplicatedStorage.CheckIn1.Value == false then    
        game.Workspace.CheckIn1.SurfaceGui.ImageLabel.Image = FirstImage
        wait(4)
        game.Workspace.CheckIn1.SurfaceGui.ImageLabel.Image = SecondImage
        wait(4) 
    elseif game.ReplicatedStorage.CheckIn1.Value == true then
         game.Workspace.CheckIn1.SurfaceGui.ImageLabel.Image = FirstImage   
        wait(4) 
        game.Workspace.CheckIn1.SurfaceGui.ImageLabel.Image = ThirdImage    
        wait(7) 
    end
wait()
end 

BUT, just FYI...

Value object changes don't replicate from client to server, so if this is a server script, the only way you can change the Value of CheckIn1 for everyone to see is from a server script or the developer console command line (F9 console). If you set it from a LocalScript, only that player's client will see things change. I mention this because having your ValueObject in ReplicatedStorage suggests you might be trying to use it to replicate a change from client to server, which will not work. You need a RemoteEvent or RemoteFunction to set it from a client and have it change on the server.