The short answer
To violate the first law of thermodynamics, Big Bang theory would have to posit that the net energy of the universe was different at two different times and that the universe always has been a closed system. Big Bang theory, however, posits neither of these.
The longer answer
The first law of thermodynamics requires the total energy of a closed system to remain constant over time (with a tiny room for error allowed by the quantum mechanical uncertainty principle, but that is not important here). For a theory about the universe to violate the first law of thermodynamics, the theory must stipulate that the universe is a closed system and that there are two distinct points in time (call them \(T_1\) and \(T_2\)) such that the total energy of the universe at \(T_1\) and the total energy of the universe at \(T_2\) do not match one another. The creationist argument implicitly assumes that Big Bang theory stipulates a time \( t < 0 \) before the origin of the universe at which time there was no energy at all (because the universe did not exist); the argument then contrasts this with the origin of the universe, after which it presumes that there has been a positive net balance of energy (because the universe exists).
We can consider this problem from both the perspective of general relativity alone and the perspective of quantum gravity. You may wish first to establish some background by reading the first section of my reply to the assertion that Big Bang theory must be false, because something cannot come from nothing.
I. No violation using general relativity alone
If we use general relativity alone, then the origin of the universe is also the origin of spacetime. That means that on this view there can be no such thing as a time \( t < 0 \). But if there is no time \( t < 0 \) when there was no energy, then the presence of any amount of energy at \( t = 0 \) cannot violate the first law of thermodynamics. I would like to interject that it is very odd that creationists would think that virtually the entire physics community would unwittingly come to accept a theory that violates one of the fundamental laws of physics.
II. No violation using quantum gravity
If we add quantum mechanics to the picture, then we are faced with Lee Smolin’s three options covered in the linked article above: (A) there is still a first moment in time; (B) the universe continues indefinitely into the past; (C) time breaks down in the very early universe. Scenario A does not violate the first law of thermodynamics for the reasons discussed above. Scenario B does not violate the first law of thermodynamics as long as the total energy of the universe remains constant at all times; unsurprisingly, models that actually fall under scenario B respect the first law of thermodynamics. I am still unclear about exactly how scenario C is supposed to work, but if there is no succession of time in the early the universe, then it would seem that the first law of thermodynamics cannot apply to the early universe, either.
III. No violation if the universe is not a closed system
If the universe came from something else (if, for instance, it started as a region of spacetime pinching off from a black hole in another spacetime, as proposed by Smolin 1997) then there still would be no conflict with the first law of thermodynamics, because on such a scenario the universe was not always a closed system, and would have inherited its initial energy from whatever it came from.
IV. The universe may have a net energy of zero
Even proposals like that of Tryon (1973), that have the universe appearing spontaneously in an empty spacetime, would not violate the first law of thermodynamics if the universe has an overall energy of zero. Strange as it may sound for the universe to have zero total energy, it may be true: the gravitational energy of the universe is a negative quantity, which may balance the rest of the energy in the universe, leaving a net balance of zero energy. This would occur in a so-called “flat universe.” As physicist Lawrence Krauss explains:
In terms of general relativity, the curvature of our expanding universe is related to the total gravitational energy of the objects being carried along with its expansion. In a flat universe, the total energy is zero. So a flat universe could have arisen from nothing. One can trade off the positive energy of particles for the negative energy of gravity and move from a situation in which there are no particles to one with a lot. (Krauss 2008)
As it turns out, results from the Wilkinson Microwave Anisotropy Probe (WMAP) have confirmed that “the universe is flat with only a 2% margin of error” (see linked site).
Krauss L. 2008. The free lunch that made our universe. New Scientist 2683:53
Smolin L. 1997. The Life of the Cosmos. Oxford: Oxford University Press.
Tryon EP. 1973. Is the universe a vacuum fluctuation? Nature 246: 396-397. https://doi.org/10.1038/246396a0