Why isn't my fade in and fade out script working for this GUI?

You're trying to set a variable to a property, then change the property from that variable on lines 37 and 44. THIS WILL NOT WORK!. The variable becomes static, not dynamic!

You have to define the variables as the object, then index the property from the variable.

function fadeIn() -- Function used to fade the text in
    mutetext.TextTransparency  = muteText.Tansparency - transparencyConstant
    mutetext.TextTransparency = math.max(0, muteText.Transparency)
end

function fadeOut()
    mutetext.TextTransparency  = muteText.Tansparency + transparencyConstant
    mutetext.TextTransparency = math.max(0, muteText.Transparency)
end

1. Don't redirect values to properties, but for objects:

-- the rest of the script

transparencyConstant = 1/15 -- The increment in which the text should fade in or out

function fadeIn() -- Function used to fade the text in
    local MTtransparency = mutetext
    MTtransparency = MTtransparency.TextTransparency - transparencyConstant -- Supposed to make less transparent (fade in) by subtracting 1/15 every wait()
    MTtransparency.TextTransparency = math.max(0, MTtransparency.TextTransparency) -- The transparency of mute text cannot be lower than 0 
    print(MTtransparency) -- Used for debugging purposes
end

function fadeOut()
    local MTtransparency = mutetext
    print(MTtransparency.TextTransparency)
    MTtransparency.TextTransparency = MTtransparency.TextTransparency + transparencyConstant -- Supposed to make more transparent by adding 1/15 every wait()
    MTtransparency.TextTransparency = math.min(1, MTtransparency.TextTransparency) -- The transparency of mute text cannot be more than 1
    print(MTtransparency) -- Used for debugging purposes
end

--the rest of the script

2. Where is the loop? It will just run once!!! (I recommend using a for loop)

-- the rest of the script, dont need constant

function fadeIn() -- Function used to fade the text in
    for i = 14, 0 , -1 do
        mutetext.TextTransparency = i/15
        mutetext.TextTransparency = math.max(0, mutetext.TextTransparency) -- The transparency of mute text cannot be lower than 0 
          print(mutetext.TextTransparency) -- Used for debugging purposes
        wait()
    end
end

function fadeIn() -- Function used to fade the text in
    for i = 1,15 do
        mutetext.TextTransparency = i/15
        mutetext.TextTransparency = math.min(0, mutetext.TextTransparency) -- The transparency of mute text cannot be lower than 0 
          print(mutetext.TextTransparency) -- Used for debugging purposes
        wait()
    end
end

--the rest of the script

And i think its just that. Hope it helped.

WHOLE SCRIPT

local mute = Instance.new("ImageButton")
mute.Name = "mute"
mute.Parent = screenGui
mute.Image = "http://www.roblox.com/asset/?id=154834667"
mute.ZIndex = 2
mute.BackgroundTransparency = 1
mute.Size = UDim2.new(0.025,0,0.0625625,0)
mute.Position = UDim2.new(0.005,0,0.9335,0)


local mutetext = Instance.new("TextLabel",screenGui)
mutetext.Name = "mutetext"
mutetext.Position = UDim2.new(0.005,0,0.91,0)
mutetext.BackgroundTransparency = 1
mutetext.TextXAlignment = "Left"
mutetext.Text = "Scroll/click to adjust music volume"
mutetext.TextWrapped = false
mutetext.TextTransparency = 1
mutetext.TextStrokeTransparency = 1
mutetext.TextScaled = false
mutetext.Visible = true
mutetext.Font = "Arial"
mutetext.FontSize = Enum.FontSize.Size14
mutetext.Active = true

mute.MouseEnter:connect(function() 
    inText = true -- Variable set to true to detect when the mouse has entered the mute icon
end)

mute.MouseLeave:connect(function()
    inText = false -- Variable set to false to detect when the mouse has left the mute icon
end)

transparencyConstant = 1/15 -- The increment in which the text should fade in or out

function fadeIn() -- Function used to fade the text in
    for i = 14, 0 , -1 do
        mutetext.TextTransparency = i/15
        mutetext.TextTransparency = math.max(0, mutetext.TextTransparency) -- The transparency of mute text cannot be lower than 0 
          print(mutetext.TextTransparency) -- Used for debugging purposes
        wait()
    end
end

function fadeIn() -- Function used to fade the text in
    for i = 1,15 do
        mutetext.TextTransparency = i/15
        mutetext.TextTransparency = math.min(0, mutetext.TextTransparency) -- The transparency of mute text cannot be lower than 0 
          print(mutetext.TextTransparency) -- Used for debugging purposes
        wait()
    end
end


local fadeInandOut = coroutine.wrap(function() -- Coroutine so that other code after it can work; other code can run simultaneously 
    while wait() do -- Checks every wait()
        if inText == true then -- If the mouse is in the mute icon, inText is true. And if inText is true, then
            fadeIn() -- (cont.) fade the mute text in.
        elseif inText == false then -- If the mouse isn't in the mute icon, inText is false. And if inText is false, then
            fadeOut() -- (cont.) fade the mute text out. 
        end
    end 
end)

fadeInandOut() -- Calls the function