Do the MHD equations break down under certain circumstances (and possible connection to fusion experiments)? The Euler equations produce singularities (in finite time) under certain conditions see here. 
When rings of fluid approach each other,  in certain simplified scenarios,  the Euler equations that describe their motion break down. 
Nevertheless,  even in more realistic (non-idealized) models, although singularities will not appear,  regions of extremely high pressure (density, vorticity) will emerge.
Question.  Could this work be extended for the MHD equations governing the behavior of plasmas? In simplistic terms, could two (high density) rings of plasma colliding lead to the emergence of regions of high density (temperature, pressure) plasma, that could facilitate fusion in a tokamak (or stellarator),  for example? See comments for (heuristic) details. 
 A: 
Do the MHD equations break down under certain circumstances?

Yes, all the time.  MHD is just a fluid approximation for a plasma, as I mention at https://physics.stackexchange.com/a/445307/59023 and https://physics.stackexchange.com/a/350655/59023 and https://physics.stackexchange.com/a/239769/59023 and https://physics.stackexchange.com/a/136592/59023.
There are multiple versions of MHD:


*

*Ideal:  infinitely conductive, magnetized fluid, and Ohm's law is just $\mathbf{E} = -\mathbf{v} \times \mathbf{B}$

*Hall:  allows for $\mathbf{j} \times \mathbf{B}$ term in Ohm's law

*Two-fluid:  treats the fluid as two separate populations and adds pressure gradient terms to Ohm's law

*several others that are just adding extra terms to Ohm's law or including extra corrections.


None of these are exact by any means, so of course trying to model reality will break down.  The question is by how much and whether the break down even matters.  MHD is incredibly good at modeling things that are intrinsically boring, e.g., homogenous and uniform, stationary magnetized fluids.  It is annoying good at modeling things that are not boring as well.  I say annoying, because it does much better than one would think it should.  However, it fails completely with collisionless shocks.
You may notice that the above link goes to an MHD article about magnetized shocks and discontinuities.  MHD is fine to describe the asymptotic state of these things, but it misses several critical features and cannot explain how the shock actually initiates or dissipates energy.  MHD is like a black box and if you provide the "correct" inputs and outputs, it can do some things quite well.  Also note that ideal MHD is scaleless.  That is, there are no relevant spatial or time scales.  In the higher order versions of MHD one can calculate these scales, but they are not physically meaningful in the impact/effect that they actually have on plasmas.
Thus, a collisionless shock, for instance, is known to have a ramp thickness ranging from a few electron inertial lengths, $c/\omega_{pe}$, where $\omega_{ps}$ is the plasma frequency of species $s$ and $c$ is the speed of light in vacuum, to an ion inertial length, $c/\omega_{pi}$.  MHD cannot handle this or even acknowledge it.  The Rankine-Hugoniot conservation relations, for instance, assume the ramp is some thin region (depending on the book you read, some even say it's infinitely thin though I find this inappropriate), where thin is defined as a scale small enough such that normal particle dynamics cannot occur.  That is, the change occurs faster than the particles can respond.  As noted above, when the ramp is smaller than the ion scales, these particles cannot undergo a complete gyro orbit within the ramp.

In simplistic terms, could two (high density) rings of plasma colliding lead to the emergence of regions of high density (temperature, pressure) plasma, that could facilitate fusion in a tokamak (or stellarator), for example?

I am not sure I follow, but coronal mass ejections often overtake previously launched ones forming shock-shock interaction regions.
As an aside, although high energies are necessary to get particles past their own Coulomb potentials, high energies also make the plasma more likely to escape.  That is, in the case of fusion experiments, the biggest obstacle is and has always been confinement.  Confinement just refers to keeping the plasma held in place long enough for enough particles to undergo fusion that the energy output meets or exceeds the energy input used to generate these conditions, also called the break even point.
