Problem with TeamColor?

Your problem is that you're trying to compare a string value to a BrickColor value. These are both different datatypes, so to use them together you need to translate one to the other. To do this, we'll simply change the string into a BrickColor.

Ex.

brickcolor = BrickColor.new("Crimson") --Sets  BrickColor value
string = "Crimson" --Sets a string value

Ok, now to apply this to your script, we'll change the string to a BrickColor like used in the example above:

game.ReplicatedStorage.Remotes.Shoot.OnServerEvent:connect(function(player, hit, damage)
    if hit then
    if hit.Parent:FindFirstChild("Humanoid") then
        if hit.Name == "Head" then
            hit.Parent.Humanoid:TakeDamage(damage * 2)
        else
            hit.Parent.Humanoid:TakeDamage(damage)
        end
    elseif hit.Parent.Parent:FindFirstChild("Humanoid") and not hit.Parent:IsA("Tool") then
        if hit.Name == "Handle" then
            hit.Parent.Parent.Humanoid:TakeDamage(damage * 2)
        else
            hit.Parent.Parent.Humanoid:TakeDamage(damage)
        end
    end
    if hit.Parent:FindFirstChild("LastHit") then
        hit.Parent.LastHit.Value = player
    elseif hit.Parent.Parent:FindFirstChild("LastHit") then
        hit.Parent.Parent.LastHit.Value = player
    end
    if hit.Parent == game.Workspace.Core and player.TeamColor == BrickColor.new("Crimson") then
        hit.Parent.Health.Value = hit.Parent.Health.Value - damage
        hit.Parent.LastHit.Value = player
    end
    end
end)

game.ReplicatedStorage.Remotes.ChangeWalkSpeed.OnServerEvent:connect(function(player, speed)
    if player.Character and player.Character.Humanoid then
        player.Character.Humanoid.WalkSpeed = speed
    end
end)

Ok, well now the code should work correctly! Anyways, if you have any further problems/questions, please leave a comment below, and I'll see what I can do. Hope I helped :P

-Dyler3