What is the origin of the pluralization "virii"?

I suspect the confusion arises out of the fact that very few people now have any kind of understanding of Latin. Many words get turned into faux-Latin plurals, even when the word itself is not Latin in origin, or when the plural form is completely incorrect (octopi, platypi, penii). It seems to come up most often when the plural form of the word has an -ses construction. It feels uncomfortable to say, so people (often jokingly, in my experience, but occasionally earnestly) try to give it a faux-Latin plural.

On the Italian Wikipedia page for Virus, it seems that there are different plurals used in different contexts.

I can't quote it because it's italian and almost no-one will understand so I'll try to translate the important parts:

It should be singularia tantum (=it only has the singular form), like "rice" or "air".

Lwoff, Horne and Tournier, within their classification in 1962, proposed and used the form "vira". In the anglophone area of interest, virii (from vīriī) is rather used referring to Computer viruses, while within biological area the one used is "viruses".

Vīriī is wrong because it would come from a word *vīrius (such as radius, radiī) which does not exist.
Another plural could be vīrī which instead would require the singular form "vir" that means man and whose plural form is "vĭrī".