[adamcharnock]

Well yes and no.

Yes you can form a company in Ireland while living in France. But you cannot get an Irish VAT number without a physical presence in Ireland.

And if – for example – France learns that you are running an Irish company from France (i.e. you have a 'permanent establishment' in France), they'll want you to file and pay French corporation tax. Which is likely sufficiently annoying that you may as well have formed a French company in the first place.