Can a system in its ground state tunnel to a state of higher energy I have heard that quantum systems with higher energy than their ground states can tunnel and fall into a lower energy state. Can the reverse happen, quantum systems tunneling to a higher energy state?
 A: You need to be a bit cautious with statements like:

tunnel and fall into a lower energy state

because energy is always conserved, so in any change the total energy of the final state is the same as the total energy of the initial state.
Suppose we take as an example a radioactive nucleus that decays by alpha decay, which is one of the classic examples where a system has to tunnel through a potential barrier in order to decay. If we just consider the nucleus than its energy is lowered by the decay so it is tempting to say that it tunnels through the barrier to a lower energy state. But the energy doesn't just disappear of course - it goes into the energy of the emitted alpha particle. So when you take the system as a whole it doesn't tunnel into a lower energy state, it tunnels into a state with the same energy.
The equations that describe the decay are time symmetric, so the reverse process is described by the same equations as the decay process. That means it's perfectly possible for an alpha particle colliding with the decayed nucleus to tunnel back into the undecayed state.
So in this sense yes a system can tunnel into a higher energy state, but as before this statement makes sense only if you are considering just the nucleus and ignoring the energy of the incoming alpha particle. Regardless of which way the system is tunnelling through the barrier the total energy remains the same.
In practice we only ever see the decay and we don't see alpha particles tunnelling back into the nucleus to undecay it. But this is simply because once the nucleus has decayed the probability of an alpha particle hitting it in just the right way to undecay it is exceedingly small. It's not the that reverse process is impossible, just that it's very unlikely.
A: Look at this tunneling diagram, it says it all.


As a particle approaches the barrier, it is described by a free particle wavefunction. When it reaches the barrier, it must satisfy the Schrodinger equation 



I have heard that quantum systems with higher energy than their ground states can tunnel and fall into a lower energy state.

After the barrier, whether there is a lower energy state it can decay to or not depends on the particular situation. It is a matter of conservation of energy that the particle cannot go to a higher energy state.
A: First of all, if we consider Schrödinger equation and our initial state is an eigenstate (it doesn't matter ground or excited) then the system stays in this state during time evolution. "Fall into a lower energy state" happens under interaction with outer fields and that is why you can't describe your system by Schrödinger equation anymore.
So, to summarise: for closed system - no (Schrödinger equation leaves your initial eigenstate) and for open systems - yes, it is possible (for example, when atom absorbs photons with appropriate frequency). But "tunneling" is not an appropriate word, I would say transition between states.
