How to fix the Overhead it suddenly stop working?

There were a few issues in your code,

In the original code, the Left and Right variables were being defined outside the Player.CharacterAdded event handler, which means that they would be defined regardless of whether the player's character has been added or not. Since these variables are used later on in the code, this could cause errors if the player's character has not been added yet.

By moving the declarations of Left and Right inside the Player.CharacterAdded event handler, we ensure that these variables are only defined when the player's character is added, which eliminates the potential for errors.

Here is a fixed version of your code:

local ReplicatedStorage = game:GetService("ReplicatedStorage")
local Players = game:GetService("Players")

local Overheads = ReplicatedStorage:WaitForChild("OverheadsGui")
local GroupId = 11540689    ----// GROUP ID

Players.PlayerAdded:Connect(function(Player)
    Player.CharacterAdded:Connect(function(Character)
        -- These variables are used regardless of whether the user owns the gamepass or not
        -- so we can define them outside the if statement
        local Humanoid = Character:WaitForChild("Humanoid")
        local Head = Character:WaitForChild("Head")
        local Left = Overheads:WaitForChild("Left")
        local Right = Overheads:WaitForChild("Right")

        local Stats = Player:WaitForChild("playerstats").Rank.Value

        -- Again, this happens regardless of the if statement
        Humanoid.DisplayDistanceType = Enum.HumanoidDisplayDistanceType.None
        if Stats == "Unrank" then
            Left.Image = "rbxassetid://11260847774"
            Right.Image = "rbxassetid://11260873360"

        elseif Stats == "Bronze" then
            Left.Image = "rbxassetid://11260880194"
            Right.Image = "rbxassetid://11260884454"

        elseif Stats == "Silver" then
            Left.Image = "rbxassetid://11260890595"
            Right.Image = "rbxassetid://11260895668"

        elseif Stats == "Gold" then
            Left.Image = "rbxassetid://11260902279"
            Right.Image = "rbxassetid://11260906114"

        elseif Stats == "GoldPlus" then
            Left.Image = "rbxassetid://11260911548"
            Right.Image = "rbxassetid://11260917532"

        elseif Stats == "Creator" then
            Left.Image = "rbxassetid://11260911548"
            Right.Image = "rbxassetid://11260917532"

        elseif Stats == "Tester" then
            Left.Image = "rbxassetid://11260911548"
            Right.Image = "rbxassetid://11260917532"

        elseif Stats == "Contributor" then
            Left.Image = "rbxassetid://11260911548"
            Right.Image = "rbxassetid://11260917532"
        end

        local CloneGui = Overheads:Clone()
        CloneGui.Username.Text = Player.Name
        CloneGui.Parent = Head
        CloneGui.Name = "OverheadGui"
        local Role = "Guest"
        pcall(function()
            Role = Player:GetRoleInGroup(GroupId)
        end)
        CloneGui.Role.Text = Role
    end)
end)