How Can I Make The Group Door Work As Intended?

You're on the right track, all you need to do is to do something to the player if they aren't in the group. This could be teleport them away from the door or simply killing them, which is the simpler option. For now we'll make it so the player dies, but you can change it as you wish.

GroupId = 3366998

script.Parent.Touched:Connect(function(hit)
	local plr = game.Players:GetPlayerFromCharacter(hit.Parent)
	if plr:IsInGroup(GroupId) then
		script.Parent.Transparency = 0.7
		script.Parent.CanCollide = false
		wait(2)
		script.Parent.Transparency = 0.15
		script.Parent.CanCollide = true
	else --If the player is not in the group then execute the code below
		local humanoid = hit.Parent:FindFirstChild("Humanoid")
		if humanoid then --Check if their humanoid exists
			humanoid.Health = 0 --If it does, then kill them.
		end
    end
end)

Problem solved! If you wanted to make the code even better now, you could leave the door >CanCollide = false as it is no longer necessary.

GroupId = 3366998

script.Parent.Touched:Connect(function(hit)
	local plr = game.Players:GetPlayerFromCharacter(hit.Parent)
	if plr:IsInGroup(GroupId) then
		script.Parent.Transparency = 0.7
		wait(2)
		script.Parent.Transparency = 0.15
	else
		local humanoid = hit.Parent:FindFirstChild("Humanoid")
		if humanoid then
			humanoid.Health = 0
		end
    end
end)

et voila!