How does Buddhism refute the inherent existence of sub-atomic particles? (1) According to Rob Burbea in Seeing That Frees, "both the neither-one-nor-many reasoning and the sevenfold reasoning are among the practices capable of revealing the emptiness of matter even at the most basic levels." 

Regarding the neither-one-nor-many practice, he goes on to say: 
"Even if we imagine down to the level of subatomic particles, these will necessarily have parts facing in different directions, or interacting with other particles in this or that direction. Anything that occupies space must have parts. 

Postulating the existence of a partless particle that would be truly singular will not work. It would be impossible to arrange or amass such particles in order to form any thing from them. Having no differentiable sides, other particles could not be arranged either side of it. Such a particle would not be able to bond or interact with other particles in any direction. All the surrounding particles would contact the central particle at the same point, and all effectively occupy the same space. Nothing with any extension could ever come to be."

When I imagine a partless particle, I simply think of a sphere. I can't see how the reasons he states refute that possibility. He seems to think that a partless particle could not possess mass, or would just be an infinitely small point. Why could it not just be a sphere that can't be divided any further?

(2) Two of the quotations he uses:

"The element of earth has no nature of its own." (Prajnaparamita Sutra)

"Matter itself is void. Voidness does not result from the destruction of matter, but the nature of matter is itself voidness." (Vimalkirtinirdesa Sutra)

What are the other "practices capable of revealing the emptiness of matter even at the most basic levels?" Specifically this belief in the inherent existence of the "element of earth", i.e. sub-atomic particles.
In this paragraph:


  Postulating the existence of a partless particle that would be truly
  singular will not work. It would be impossible to ... Having no
  differentiable sides, ... Such a particle would not be able to ... All
  the surrounding particles would ... Nothing with any extension could
  ever come to be.


The author is, in my estimation (I have no familiarity with the book or the author), doing one of two things:


Confusing Buddhism with Physics. Whether this particle is this or that particle is that, it won't help you become a better person and understand your mental reality any better. Although the above quoted paragraph would make an interesting question on physics.stackexchange.com.
Speaking poetically in order to help the reader become absorbed with the idea. Perhaps this is the case.


During an course I attended under the name "Introduction to Asian Philosophy", my professor explained emptiness as a flashy way of reminding us that we only define things in relation to what they are not. Nirvana is defined as everything Samsara is not, therefore Nirvana is "empty of essential qualities". Similarly, Samsara is meaningless without a Nirvana to compare it to. This logic can be extended to all entities and concepts: Without night there can be no day, without a whole there can't be parts, etc.

You may or may or not you agree with this reasoning, but for me it's a practical interpretation of the doctrine of emptiness that helps us to not cling to our perceptions of what things are.

In other words, as R. Barzell says in the comment below, Is emptiness ontology, or a way of seeing to free us from the bondage of concepts? Is treating emptiness as ontology missing the point?

Lastly,


  What are the other "practices capable of revealing the emptiness of matter even at the most basic levels?" Specifically this belief in the inherent existence of the "element of earth", i.e. sub-atomic particles.


Meditation will probably help, as it gradually helps the practitioner become free from the limitations of language, thoughts, and concepts. The better we understand the way our minds work, the less we can be obstructed by our minds.