Does the Rabbi think this experiment is evidence for the existence of a metaphysical dimension through which information can be transmitted?

I don’t know. Note that even defining it as metaphysical is problematic. It could also be some other sense beyond those familiar to us. For example, animals perceive ranges of sight and hearing beyond those that we perceive.

For the sake of discussion, if there were an experiment showing transmission of information between two people over a very large distance while both are in a room sealed against sound, smell, electromagnetic fields, etc. — would that constitute evidence for the existence of a metaphysical dimension through which information can be transmitted?

I don’t understand why you need the assumption of a great distance if there is complete sealing. To exceed the speed of light?
As for the question itself, again, it’s hard to answer. What does metaphysical mean? Maybe it’s another sense? If it’s outside the light cone, then maybe, because no information can pass that fast. But beyond that, there is always the possibility of some mistake (the sealing isn’t complete, and so on).
I wrote about this difficulty in my book God Plays Dice, regarding Dawkins’ claim that he doesn’t believe in anything metaphysical. That’s a fairly empty statement in terms of content (certainly empirical content).

The distance is meant to ensure that no information is being transmitted over a physical range of influence (on the assumption that the greater the distance, the harder any physical influence becomes to detect).

As for what you said, that maybe it’s another sense — what other sense could be relevant under such conditions? And beyond that, if we have a satisfactory explanation (a metaphysical sense), why assume that maybe there exists some other sense responsible for receiving the information? That sounds to me like a skeptical claim.

I just saw that the academic researcher Rupert Sheldrake (who has already been mentioned here on the site in the past) wrote a book about this matter of dogs’ telepathic ability. The title of the book is Dogs That Know When Their Owners Are Coming Home.
In this short video he explains the phenomenon a bit:
https://www.youtube.com/watch?v=9QsPWitQovM

If you’re talking about a practical question, there is always the possibility that we missed something. The distance makes no difference. The question is hypothetical: suppose it were clear to us that there is no natural process, meaning absolute sealing. The distance isn’t important.
The question of what counts as metaphysical is a matter of definition. If everything familiar to us is physical, and everything beyond that is metaphysical, then indeed when we discover something unfamiliar we will classify it as metaphysical. But I don’t think that’s a good definition (although I don’t know a better one, and therefore in my view the question is not well-defined). It’s like the debate over mystical medicine and resorting to powers from the “other side.” According to Nachmanides this is forbidden, while for Maimonides there is no such thing (and therefore if it works, it isn’t from the “other side”). The question that arises according to Nachmanides is: what is the criterion for determining that some power comes from one “side” or another? For Maimonides the criterion is that it works, but for Nachmanides it’s unclear. The same applies here.
Sheldrake is a highly controversial researcher (see, for example, his shifting value of the gravitational constant). In any case, for our purposes I’m still waiting for a definition of what a metaphysical matter is.

As far as I remember, you’re a dualist. That is, you acknowledge the existence of a soul. A soul is an example of something that belongs to the metaphysical dimension. Our consciousness probably belongs to that dimension too.

I am indeed a dualist. But the question here was how one could prove dualism scientifically, and my answer is that one cannot. This is a philosophical topic, not a scientific one. I assume there is a non-material dimension within us. But any practical difference you formulate for a laboratory experiment can also be explained without that — unless you discover an electron moving without any force acting on it (solely by the power of will). And even then one could always wonder whether there was some force there that we didn’t measure.

By the way, the Holy One, blessed be He, also belongs to the metaphysical dimension, and indeed with regard to Him too there is no possibility of measuring and proving His existence by experiment.

https://youtu.be/qX1-Hth4HEw
It’s worth watching the end of the video about the amazing results of the dog study. I can’t say I was surprised.