The name Assyria comes from the first Assyrian capital, the city of Ashur on the upper Tigris, and the name Asura comes from the Proto-Indo-European root h2-ens-u-s, more or less meaning one or more powerful spirits(compare Avestan Ahura Mazda, "Wise Lord", Old Norse Aesir, etc). So no, there is most likely no relationship beyond a very vaguely similar name.

Not sure about the Assyrians, but you'd probably be interested in the asuras and devas of Zoroastrianism.

They are not the same as the Assyrian people. However, they are not unrelated. Both Indian and Zoroastrian mythology have beings referred to as Asuras and Devas (Ahuras and Daevas in Zoroastrianism). In Indian mythology, Devas are just deities. The word is etymologically rooted in Indo-European deiwos, which is also the root for deity, Jupiter (a foreshortening of deiwos pater, sky/divine father) and Zeus. Basically they were sky and nature deities, though later elaborated. Asuras (contra to Suras) were lesser deities. Over time the word acquired a negative connotation, eventually becoming equivalent to demons.

In ancient Iran, the words wound up being used in the opposite ways, and the Ahuras, most prominently Ahura Mazda, became viewed as the supreme upholder of Truth, while the Daevas were deceitful proponents of "the lie", which interestingly has a cognate concept in Indian religion called Maya, which is sort of the veil of illusion.

Curiously the Aesir of Norse mythology come from the same etymological roots as the Asura.

Don't think so, what would lead you to believe that other than the similar name?