Suppose we have a steady state universe with a gas chamber resembling that of Maxwell's demon that is used to power this hypothetical heat engine as molecules transfer to their respectable sides based on temperature. Now, suppose we ran this machine for infinity where eventually it reaches a thermodynamic equilibrium. However, every so often, Schrodinger's wave equation could allow one of these molecules to switch sides, hence powering the engine as it again transfers to its respectable side. What flaw in this experiment prevents this machine from becoming a perpetual motion machine and breaking the 2nd law of thermodynamics as the molecules transfer back and forth every so often for infinity?
Firstly, the assumption that you could build a working Maxwell Daemon (a three state machine with Szilard-engine actuation hardware) to extract work from the system already gainsays the assumption of a steady state universe.
The problems of mixing classical and quantum statistical mechanics aside, the tunnelling here is in principle no different from classical translational motion through the door operated by the classical Maxwell Daemon. Here, the Daemon would work by waiting until it knew there was a random fluctuation making one side of the wall hotter. Then it could do its stuff - through a Szilard engine.
But then the standard objections to the Maxwell Daemon would apply. See my answer here for more details. The Daemon itself can and does work - we've actually built one in the laboratory - the problem is that it must do measurements to work and if it is a microscopically reversible system (look up Loschmidt's paradox), it thus transfers the entropy of the gas states it is watching to its computer memory. When this memory fills up, it must erase it, a process which is also microscopically reversible, then this entropy must be transferred to the states of the external system - now we're back to square one.
The entropy of all theromodynamic systems fluctuates up and down, with the size of the fluctuations inversely proportional to the size of the number of particles so you would indeed see local, short lived fluctuations. Indeed, over longer and longer times, you would eventually see bigger and bigger imbalances purely by chance. This line of reasoning leads to the Boltzmann Brain problem - look this up.
To answer this I need first to clarify what the scenario is. I think it is the following. A gas is in two parts of a vessel, and the two parts have reached thermal equilibrium with one another so no heat engine can operate between them. However, every now and then a molecule will change sides, whether by tunneling or another process. The two sides are then slightly out of equilibrium (says the argument) so now a heat engine can run and extract a little energy, returning them to equilibrium, and now the whole process can repeat ad infinitum. The gas slowly cools overall, and useful work is extracted. This would break the 2nd law if it happened.
The flaw in the reasoning here is not essentially different from the flaw in applying this same reasoning to two parts of an ordinary gas in a single chamber. The transfer of molecules between sides is happening all the time by thermal fluctuations, so people have suggested that one could simply wait for a suitable thermal fluctuation, and then run an engine to exploit it. There's a nice analysis of a ratchet working on this principle in the Feynman lectures on physics. One finds that in fact it acts just like any other heat engine. In order to exploit a thermal fluctuation it needs to have access to another region at a lower temperature, and it exports heat to that other region in just such a way that entropy overall is conserved (if we assume the whole apparatus is reversible; if it is not then the efficiency is worse just as Carnot's theorem says).
The case of a Maxwell daemon is one where the heat engine works through an intermediate stage in which information is stored internally. However, eventually the internal memory has to be cleared and this is the step which unavoidably involves in increase of entropy in the environment (as a previous answer correctly asserted). As I understand it, Landauer proposed this resolution and Bennett worked it out more thoroughly.
To conclude, then, you can indeed extract mechanical work by exploiting a thermal (or other type of) fluctuation in some system A, where A is in internal equilibrium, but only by having access to system B whose temperature is lower than that of A. If the process acts reversibly then one finds that the heats exchanged at the two places A and B are in proportion to $T_A/T_B$ so it achieves the same efficiency that the Carnot cycle achieves.