Q:

Why is quantum tunneling most significant between states of equal energy?

Why is it, that quantum tunneling is only significant between states of nearly equal energy (as claimed here: 'Since tunnelling is significant only between states of nearly equal energy, tunnelling is unlikely in such instances.')?

A:

First, to answer your question. Tunneling in Quantum Mechanics is, of course, about the transfer of something through a potential barrier, which would be impossible in classical mechanics. 
The rate of tunneling depends strongly on the mass on the particle, the height of the potential barrier and the width of the barrier.
The paper that you mention in your question describes how enzymes can make reactions go faster by increasing the rate of tunneling for electrons and protons (hydrogen nuclei - or in effect ionized H atoms). The paper suggest that the key mechanism is by reducing the width of potential energy barriers. 
For an electron or proton to tunnel through a barrier and remain on the far side of it (and not return) it should have a quantum state to fit into on the other side of the barrier. If there is not a quantum state with similar energy on the other side of the barrier then the probabiliy of the electron or proton being located on the far side of the barrier becomes very small. (Strictly speaking there will be individual quantum states that span space on both sides of the potential energy barrier). If there is an available state with a different energy on the other side of the barrier then, as well as tunneling, energy would have to be gained or lost (possibly by photon absorption or emission) to allow the particle to get to the other side of the barrier and the probability for this is very low. 
Finally from your comment, you are interested in magnetization. I am not sure that I understand how magnetization relates to tunneling, unless proton/electron transfer from molecule to molecule is important. 
Edit more information about two states with similar energy each side....

[Figure taken from http://www.scielo.br/img/fbpe/jbchs/v08n5/a19fig04.gif is about restricted motion inside a molecule.]
In the figure below the top of the barrier in the centre, there would be a single state on each side of the barrier, but these two states (one on the left and one on the right) mix with each other to make two states that exist at the same time on both sides of the potential barrier. 
So the particle can pass from one side of the barrier to the other on a single quantum state. 
Thus if we have two states of similar energy on each side of the barrier they mix together to make two states which are on both sides of the potential and there is no need to energy loss / gain as the system changes from one side to the other

