How to stop the names above players' heads from switching colors when they change teams?

You have to do this by using some BillboardGui's and Hiding the players display name.

Server script ( put this in ServerScriptService ):

local Players = game:GetService("Players")

local fonts = {
    Fredoka = Enum.Font.FredokaOne,
    Bangers = Enum.Font.Bangers,
    Creepster = Enum.Font.Creepster,
    Arcade = Enum.Font.Arcade,
    Classic = Enum.Font.Code
}

local function Tag(Character, Text, Color, Font)

  local R = Color[1]
  local G = Color[2]
  local B = Color[3]

  if Font ~= nil then
     ChosenFont = Font
  end

  if Font == nil then
    ChosenFont = fonts.Classic
  end

  local BBGUI = Instance.new("BillboardGui")
  local BBGUI_Text = Instance.new("TextLabel")

  BBGUI.Parent = Character:FindFirstChild"Head"
  BBGUI.ZIndexBehavior = Enum.ZIndexBehavior.Sibling
  BBGUI.Active = true
  BBGUI.AlwaysOnTop = true
  BBGUI.LightInfluence = 1.000
  BBGUI.Size = UDim2.new(0, 200, 0, 50)
  BBGUI.StudsOffset = Vector3.new(0, 2, 0)

  BBGUI_Text.Parent = BBGUI
  BBGUI_Text.BackgroundColor3 = Color3.fromRGB(255, 255, 255)
  BBGUI_Text.BackgroundTransparency = 1.000
  BBGUI_Text.Size = UDim2.new(0, 200, 0, 50)
  BBGUI_Text.Font = ChosenFont
  BBGUI_Text.Text = Text
  BBGUI_Text.TextColor3 = Color3.fromRGB(R, G, B)
  BBGUI_Text.TextScaled = true
  BBGUI_Text.TextSize = 14.00
  BBGUI_Text.TextWrapped = false

end

local function CharacterAdded(Character)
      Tag(Character, Character.Name, {255,255,255}, fonts.Classic)
end

local function Hide(Player)
    Player.HealthDisplayDistance = 0
    Player.NameDisplayDistance = 0

    Player.CharacterAdded:Connect(CharacterAdde)
end

Players.PlayerAdded:Connect(Hide)