First of all, are you a tax professional, @Grantage?
I'm not, but I do have a fairly good grasp on international tax law, nonetheless. And I really struggle taking your "FACTS" seriously.
First of all, the case you refer to (Schumacker - C-279/93), does nothing to say that you only have a tax liability to the country of incorporation for the legal entity you are collecting a paycheck from.
To fill in others I'll briefly summarize what the case is about: it is about a Belgian resident (i.e. he is not "a resident of no country") physically working in Germany for a German employer (i.e. he is working for a company that is clearly tax resident in Germany) and receiving more than 90% of his income there. It was a tax treaty between Belgium and Germany that gave Germany the right to tax the full income, not EU law, and not what the case was about. The case was about Germany's right to discriminate against the Belgian national by not allowing him the same rights to deductions and tax rebates (including "splitting of income" between spouses) as a German resident. And that's where European Law was applicable, in stating that due to the free movement of labor within the European Community, Germany was not allowed to engage in such discrimination.
So to take a look at your example with a Finnish national (and assumed resident until the point where your example started). First of all, Finnish residents are deemed to be a tax resident for 3 calendar years after leaving the country. It doesn't mean that they must pay taxes to Finland in all cases, e.g. if they are resident in a country with a tax treaty with Finland, they might pay some or all of their taxes there (depending on the treaty).
If you don't understand the basics of international tax law, I can see how you might think that registering a company in a third country (with a tax treaty with Finland), and then paying yourself a salary from that company might shift your tax liability to that country. The problem with this is the typical "mind and management" test, where unless you actually become a personal tax resident of the country of incorporation (assuming you're sole owner of the company), Finland would be likely to consider the company to be a Finnish resident. So without physically working for an actual established company with actual economic activity and management control within that country, no such treaty (nor the "EU law" you apparently do not understand) will give you any benefit.
It is however possible to be a resident of no country, and have no tax liability anywhere. But that has nothing to do with EU law. You will probably be skirting some laws, but at very little risk. See Streber's excellent article on the topic. The exception to this (like Edmund aka @FlagTheory correctly stated above), is depending on the laws of the country of your previous tax residency. Or in the case of the US and Eritrea, depending on your citizenship.
@jdmcwalter I would be wary of trusting any advice on this thread (including my own), and especially the one you just got from @Grantage. Instead, talk with tax professionals in each country that might somehow want to collect taxes from you.