Are there any QM effects where charged particles are not intimately involved? Are there any QM effects that have been/could be measured from interactions involving non-charged particles? 
Elementary QM is all about the electron energy levels in the atom, photon - atom interactions, etc. 
When one looks at the nucleus, its all about quark interactions - which are also charged particles. 
I can think of some theoretical ones - like neutrinos orbiting a mass, but they would be hard to impossible to measure. Another possibility is the strong / weak nuclear force - but that always happens with particles that are also charged. 
In the end we always need matter built instruments to see a result - that's fine. You could for instance observe photons coming from some distant (metres to Mpc) away region where some interaction occurred.
 A: Am I missing something here? Photons and the double-slit experiments do the trick. Likewise, you can entangle the polarisation degrees of freedom of photons. This is inherently quantum and photons aren't charged. True, they are the excitations of electromagnetic fields, but they are not charged.
Anyway, I guess the problem here is that except for the neutrinos and certain gauge bosons, any elementary particle carries some charge, hence interacts with electromagnetic fields. However, that doesn't mean that charge is in any way special. You can work out (and "find" in condensed matter systems, I believe) a lot of different quantum field theories without electromagnetic interactions. The framework of QM is completely independent of charged particles - it's just that the electromagnetic force is pretty strong, nearly every particle participates (in contrast to the strong force, where all leptons are uncharged) and the easiest to work with in a lab that makes you feel that it is really special with respect to QM.
A: Every particle that interacts with something is charged -- in some sense. We say that particles are charged under a certain interaction if they are affected by that interaction, and their charge measures how strongly they interact.
But what you seem to be asking about is electric charge. Then one idea of an observable phenomenon could be quark confinement. It is widely held (but not yet proved) that under the strong interaction, quarks must form bound states like protons or neutrons. Presumably this fact does not rest on the quarks also having electric charge. But this is slightly cheating with respect to your question, I feel, since physicals quarks do have electric charge.
The weak interaction allows through neutrino-neutrino scattering mediated by a $Z_0$ boson. These particles are electrically neutral. But neutrinos are hard enough to detect as-is, so even though this process is out there contributing, detecting it in an experiment is most likely far away.
Edit after the answer was accepted By interpreting your "QM effects" more broadly, I would like to suggest


*

*Entaglement (from Martin's answer below)

*The Pauli principle and Bose-Einstein condensation

*Quantization of angular momentum (PhotonicBoom's answer below made me think about this)
and also

*Heisenberg's uncertainty principle


These four are valid in any quantum mechanical theory just from the basic structure of quantum mechanics. (In order they follow from the principles that: states are vectors in a Hilbert space, composite systems are described by tensor products; particles can be indistinguishable and are fermions or bosons; the rotation group [Lorentz group in relativistic theories] acts on the Hilbert space; observables do not necessarily commute.) Even non-interacting particles have to obey these principles.
At least Heisenberg's, entanglement and Bose-Einsten condensation can be observed with atoms, which are electrically neutral. Electrons in a solid can be modeled fairly well as a free electron gas, and then the Pauli principle is very important. Both these examples are a somewhat cheating still, since atoms are bound states of the electromagnetic interaction, which is also what keeps a solid together, and the electrons in the solid.
So there are features of quantum mechanics that apply to electrically neutral particles, but it is hard to come up with an example of how to observe them for electrically neutral particles, simply because most particles in our universe have electric charge. The particles that do not, like the neutrinos and the $Z_0$ boson, interact only very weakly, but in principle that is an allowed scattering involving only electrically uncharged particles.
A: Weak and Strong interactions do not involve charge. Strong interactions involve color charge, which is a different property than "normal" charge. The weak interaction mostly involves flavor change.
As to your non-charge dependent Quantum Mechanical effects, we have tunnelling. See for example this link.
