Text stops changing color after multiple texts?

Your loop changes the newchatline variable.

At first, it works fine. You set the variable equal to the first line of text, and it changes color.

So now you want to add a second line of text. But now you change the newchatline variable to equal this new line of text.

Since the loop only changes the color of newchatline, it will not change the first line of text because newchatline now equals the second line.

What you need is a loop to run for each individual chat line. You can make use of parameters and arguments for this.

function rainbowtext(gui)
    while wait(0.1) do
        --Just making it a random color is shorter and would probably be fine in your case
        gui.Color.TextColor3 = Color3.new(math.random(), math.random(), math.random()) 
    end
end

So now instead of relying on what a specific variable equals, this loop will continue to change what the parameter equals. Each time we call the function, we can make the parameter equal something different, causing separate loops to run for each line of text.

rainbowtext(newchatline)

Since newchatline constantly changes to equal the newest line of text, this will cause a loop to run for each new line. But since we stored what it equals in a parameter, the loops will still run correctly once newchatline changes.

Yo bro, You need to make sure that your loop ends! You didnt break the loop! Also the code could have been neater and very very simple so i took the pleasure of doing it for you! Change your first function and enter this instead. It works way better! Thumbs up and Like if helped!

function rainbowtext()

    wait(.1)
    newchatline.TextColor3 = Color3.new(math.random(0,255)/255, math.random(0,255)/255, math.random(0,255)/255)

end

If you are trying to change the TextColor of your text just simply do this:

function rainbowtext()
while wait() do
newchatline.TextColor3 = BrickColor.Random().Color -- OR  color3.new(math.random(), math.random(), math.random()) if you don't wanna be limited to roblox's colors (thx you Perci1)
end
end

function NewLabel(parent, msg, player)
    newchatline = Instance.new("TextLabel", parent)
    newchatline.Size = UDim2.new(1,0,0,30)
    newchatline.Position = UDim2.new(0,0,1,-30)
    newchatline.Font = "SourceSansBold"
    newchatline.TextStrokeTransparency = 0
    newchatline.BackgroundTransparency = 1
    newchatline.BorderSizePixel = 0
    newchatline.FontSize = "Size18"
    newchatline.TextXAlignment = "Left"
    newchatline.TextYAlignment = "Top"
    newchatline.ClipsDescendants = true
    newchatline.Name = "line1"
    if player.Name == "Player1" then
        newchatline.Text = "[Owner]"..player.Name..": "..msg
        rainbowtext()
    else
    newchatline.TextColor3 = Color3.new(1,1,1)
    newchatline.Text = player.Name.. ": " ..msg     
    end
end

function UpdateOldLabels(Parent)
    for i,v in pairs(Parent:GetChildren()) do
        if v.Name:sub(1,4):lower() == "line" then
            local LineNumber = v.Name:sub(5)
            if LineNumber == "5" then
                v:Destroy()
            else
                v.Name = "line"..tostring(tonumber(LineNumber) + 1)
                v.Position = v.Position - UDim2.new(0,0,0,25)
            end
        end
    end
end

game:GetService("Players").PlayerAdded:connect(function(player)
    player.Chatted:connect(function(msg)
        for _,v in ipairs(game:GetService("Players"):GetChildren()) do
            UpdateOldLabels(v:WaitForChild("PlayerGui").ScreenGui.Frame)
            UpdateOldLabels(game:GetService("StarterGui").ScreenGui.Frame)
            NewLabel(v:WaitForChild("PlayerGui").ScreenGui.Frame, msg, player)
            NewLabel(game:GetService("StarterGui").ScreenGui.Frame, msg, player)    
        end
    end)
end)

--By DZ3 this has not been tested.