In light of the now undisputed existence of the phenomenon of quantum entanglement and it's implications, can any field of science (e.g. Neuroscience and, specifically, the study of consciousness) still draw any legitimate experimental conclusions (using standard methods of empirical science) without taking into account a non-dualistic framework (i.e. non-local effects)?
It is a common misconception that science must be carried out using the most general model available. You can still use Newtonian definitions for gravity to explain why balls roll downhill and not uphill, even though Newtonian physics has been firmly supplanted by relativity. All you have to do to make a valid scientific conclusion is demonstrate that your assumptions are reasonable for the study. In the case of neuroscience, it is typically agreed that macroscopic classical mechanics is a sufficient model to build from.
One major reason the sciences can get away with this is that every one of their conclusions comes with some statistical confidence. When they announce "we found the Higgs Boson," what they really stated was "The best models that lack a Higgs boson could only produce the data we have observed 0.000001% of the time." While you don't hear this extra information in most public forums, the actual journal articles describing the discovery of the Higgs Boson are very careful to define their confidence in the result.
The probability of entanglement having an effect on a macroscopic system is astronomically small. For "macroscopic" systems, those probabilities are typically described of a probability of occurrence of 10^-x, where x is easily in the high double digits, and quickly climbs into 3 and 4 digits. In fact, the definition of "macroscopic" is often "the point where you can ignore quantum effects." Particle physics often call for 6-sigma results, meaning 99.9999% confidence in the result -- a one in a million chance of occurring due to dumb statistical luck. Medical research such as neuroscience is typically more lenient, requiring an even weaker confidence. Adding a 0.000000000000000001% chance that an entanglement behavior was the real cause is dwarfed by the statistical probabilities already present in the measurement. And I happened to choose just 10^-18 for that arbitrary probability above. In real scenarios, the probability is even lower, and thus even more dwarfed by the statistical uncertainty of measurements.
There are scientists who are exploring the possibility that results seen in neuroscience are due to quantum effects, particularly those studying consciousness. However, these scientists are few in number because they fight a daunting challenge. No one will take their claims as a valid scientific experimental conclusion without an experiment. To run an experiment, they first have to demonstrate that the potential for quantum effects to affect the brain meaningfully is much higher than anyone else believes. They can't have some 10^-100 probability that it's happening. They need to demonstrate that there are structures in the neurons whose key behaviors are poorly modeled without quantum physics. Then, once they identify such a potential structure, they need to develop an experiment which can demonstrate that these effects matter with sufficient statistical rigor to stand next to other neuroscientific theories.
There is work being done in this direction. There are some microtubules in the brain which appear to exhibit resonance at the quantum level. However, it has not received a great deal of attention because other scientists are not convinced these effects play a large enough part in neurological activity to warrant further exploration. Those studying it have to fight dearly to find funding to continue their work. Whether this is a travesty, or a natural part of the reality of science is a matter of opinion. In either case, without some clear link to the quantum realm, the statistical variances from quantum behaviors are simply insufficient to call into question the body of existing work based on classical results.
I will give a general answer on the relationship between QM and consciousness, and ignore for now the OP's ambiguous use of the terms non-duality and non-locality
Quantum Mechanics (QM) did indeed originally pose a serious challenge to both the study of consciousness (and by extension neuroscience) on one hand, and to epistemology on the other hand (the theory of knowledge). This challenge does not come directly from entanglement, from the measurement problem.
Usually the connection between QM and consciousness is not discussed from the point of view of entanglement per-se. Entanglement is however a consequence of quantum super-positions and measurements, and is thus related to the question, through it's connection to the measurement problem in general. There are some exceptions, see the Nature article mentioned at the end of the answer.
The challenge was the following: QM seems to indicate that nature follows two different set of rules, depending on whether a conscious observer is directly measuring the experiment or not. Super-positions of state (wave/particle duality) persisted only as long as no one was looking directly at the objects being studied. The moment a conscious observer entered the picture the wave function collapsed, and the particle lost its "quantum" nature. Put in more formal terms, the evolution of quantum systems was linear (hamiltonian evolution) when no one was looking and non-linear when the observer intervened.
This seemed to give the observer a privileged status compared to other physical systems. Observation requires consciousness, and there had to be something special about consciousness for it to have its own set of rules, separate from all other physical systems. To many, this was oddly similar to Substance, or Cartesian, dualism and Descartes' earlier statement that the mind had properties which were radically different from those of physical substances. Descartes concluded that mind had to be a separate type of substance, hence the term dualism (matter substance vs mind substance).
QM seemed to confirm this general concept of a fundamental difference between mind and matter (even if the details were somewhat different from DesCartes' original formulation): physical objects followed linear rules, but mental processes led to non-linear collapse, so they must be different types of entities.
The second reason why QM was of interest to philosophers of mind was that it offered a way out of the free-will dilemma: The success of modern science and especially Newtonian mechanics had many scientist believing that since the world was governed by the laws of physics, we have no free-will, as expressed most famously by Laplace. QM was very seductive from this point of view, since it offered a way out, where the universe can still be physical but leave space for non-determinism and eventually free-will.
There are several who discussed the relationship between QM and consciousness, but probably the most famous example is Roger Penrose, who along with Hamerhoff, proposed his Orch-Or model, of how quantum effects inside neurons were responsible for consciousness.
Penrose was not well received: Physicists dismissed his claims on the grounds that the brain was too warm and neurons were too big for them to work along quantum principles. Quantum computers typically operate at a much smaller scale and than human neurons and at very, very low temperatures (close to absolute zero). Philosophers of mind dismissed him, because the current fashion right now is functionalist materialism, and Penrose is a Platonist/Dualist. Computer scientists who worked on machine intelligence hated him because he believed indicated that strong AI was impossible (although his reasons had to do with more than just QM).
Many physicists now hold that the QM measurement problem has been solved and that it does not require a conscious observer to have wave function, thanks to decoherence theory, which states that it is interaction with the environment and measurement apparatus, not the observer, which causes the collapse. If this is true, then there is no relationship between QM, consciousness and neuroscience, other than perhaps indirectly through the problem of freewill. This view is not accepted by everyone though.
In fact a recent paper in Nature, provides a direct link between entanglement of theories of mind and consciousness. In it the authors argue that the results of QM, challenge not only locality, but realism as well. Realism is the position that reality has an existence independent of the observer, in opposition to idealism, which states that the mental is the most fundamental level of reality. The authors, don't use the word idealism, but come pretty close. To quote the authors:
Most working scientists hold fast to the concept of ‘realism’—a viewpoint according to which an external reality exists independent of observation. But quantum physics has shattered some of our cornerstone beliefs. According to Bell’s theorem, any theory that is based on the joint assumption of realism and locality (meaning that local events cannot be affected by actions in space-like separated regions) is at variance with certain quantum predictions. Experiments with entangled pairs of particles have amply confirmed these quantum predictions, thus rendering local realistic theories untenable.[....]Our result suggests that giving up the concept of locality is not sufficient to be consistent with quantum experiments, unless certain intuitive features of realism are abandoned.
(Can you hear Berkeley and Kant laughing from the beyond?)
It is unfortunate that the study of the relationship between consciousness and QM has gotten a bad rep because of the likes of Deepak Chopra and his pseudo-scientific non-sense, so that even established physicists like Penrose are dismissed when they bring the connection up.
This is an interesting article of why there still might to be some value in studying QM and mind.
Quantum entanglement is the fact that under certain circumstances microphysical objects (e.g., photons, electrons) originating from a common source cannot be treated as separate particles.
Instead all particles from the source keep their system properties for a certain time. Hence interaction with one particle immediately effects the system and hereby all other particles of the system (non-locality).
In general the effects need careful preparation of the experimental situation.
Quantum entanglement is an active field of research. Hence not all questions have already found their answer. In particular it still has to be investigated how ubiquitous the effect is, is it restricted to the quantum level, etc.?
I do not know about a experiment which shows that quantum entanglement is relevant for neuroscience.
When drawing a vague analogy: The quantum mechanical uncertainty relation did not show any relevance for neuroscience. Because the neuronal assemblies in question act on a much larger scale than the scale of microphysics. Possibly the same holds for quantum entanglement.
Aside, quantum entanglement is generally termed non-local. Did you also find the heading non-dualistic? Because the terms dualistic and non-dualistic have a different meaning in a philosophical context.