I don't know what precisely "nerve signals" is supposed to refer to, but neurons exchange information mainly via one pathway: neurotransmitters. And these do not travel the synaptic cleft via quantum tunnelling - obviously, since quantum tunnelling is a phenomenon on a quantum scale (concerning electrons), while neurotransmitters are far larger, at the molecule level.
There are a number of other ways by which neurons cooperate, such as ephaptic coupling. However, the main one is the release of neurotransmitters following an action potential in the presynaptic neuron. Neurons are connected at their synapses, mostly via axo-dendritic chemical ones. That means that one neuron's axon (its "output line") is met by a part of another neuron's dendrite tree. The point where the two meet is the synapse. Axon and dendrite do not actually touch; there is a microscopic "synaptic cleft" between the two, in which the extracellular fluid which fills the brain flows. I assume it is the synaptic cleft which the "dendritic synaptic barrier" is supposed to refer to.
During the action potential, the presynaptic neuron releases neurotransmitters, such as GABA or Glutamate, into the cleft. These passively diffuse within the cleft, eventually coming in contact with the post-synaptic neuron. There, they bind to receptors which activate the post-synaptic neuron.
As far as I know, neither the release of neurotransmitters at the presynaptic neuron, nor the passive diffusion process through the synaptic cleft, or the binding to postsynaptic receptors require quantum tunnelling to any interesting degree. Basically, the stuff just floats around in the extremely narrow synaptic cleft until it comes across a receptor.
Possibly, the "person" you are talking about means some other aspect of the process, such as the receptor binding. Alas, I do not see how the specific way by which action potentials propagate is relevant for consciousness. Consciousness does not seem to depend on what happens at a specific synapse, but to the large-scale patterns of neuronal activity. If neurons were to communicate via lasers, electrically, or as they do (chemically) hardly seems to matter - what matters is that they communicate, and which neurons communicate, and what they communicate.
All of this are very basic, well-established facts about the workings of the brain. The wikipedia article is a perfectly fine introduction, though literally any neuroscience textbook will explain it like that, and countless research papers in peer-reviewed journals have investigated the process in detail, e.g. this.
There exist a number of theories regarding cognitively relevant quantum phenomena in the brain, most prominently probably Orch OR. As far as I understand it, Orch OR does not focus on the communication between neurons at chemical synapses, but on quantum states within neurons and other forms of coupling (some ephaptic coupling/electrical synapse hybrid?). Regardless of what Orch OR proposes of how the brain works, the chemical synapse is commonly thought to be the primary way by which brain neurons exchange information, and there, neurotransmitters don't tunnel, but diffuse - trivially; since neurotransmitters, being molecules, are far too large to quantum tunnel.
Furthermore, Orch OR is universally assumed to be a rather bad model of the brain.
Certainly, other perspectives exist; but the basic process of action potential propagation via chemical synapses has a well-deserved near dogmatic status in the neurosciences.
Lastly, the person you're quoting seems quite confused about very fundamental brain anatomy. For example, there are no "dendrite brain cells". The brain hosts neurons, each a single cell, each with a dendrite tree (see here for a famous picture). I wouldn't expect anybody who doesn't even understand what a neuron is to revolutionise neuroscience any time soon.