First, it might be interesting to note that all this does not exactly work the way you describe in the question. Euros are not printed by the governments. They are issued by the national central banks, under the oversight of the European Central Bank, but printed by a number of printing shops, some of them that were historically government-controlled, some not. And not every country has one.
Some countries even have several, printing euros on behalf of other countries (e.g. France with the Banque de France, which prints most – but not all! – French euros and F.C. Oberthür, a private company printing euros for the likes of Slovakia and Malta; same thing for Germany with Giesecke & Devrient). Some countries exclusively use private printing shops, locally or abroad, and some use both private and government prints.
And some government prints still work only for their own national central bank (Banque de France, Fábrica Nacional de Moneda y Timbre…) while others have turned themselves into service providers printing euros for other countries too (e.g. the Österreichische Banknoten und Sicherheitsdruck prints euros for four other countries beside Austria). Stranger still, some euro banknotes are printed… in the UK, by De La Rue, on behalf of countries like Finland or Portugal.
That said, the Bank of Greece does print some of its own banknotes so, rules or process aside, they must presumably have the tools to do it somewhere in Greece. But while the Bank of Greece and the other national central banks physically issue all new banknotes, they have no authority to decide how much (cf. article 128 of the TFEU). Usurping that authority would create a major breach with the rest of the eurozone.
This also means that if the Bank of Greece prints banknotes without the approval of the ECB Governing Council, the validity of the new banknotes (which could presumably be tracked through their serial number) or perhaps of all banknotes printed in Greece (easily identified by the first letter in their serial number) would be very unclear. And if the printer reuses serial numbers or hijacks another printer's code to avoid this, it would just be counterfeiting money, without any pretence at issuing legitimate banknotes so what would be the point?
Now, to address your core question directly, what would the consequences be should it really happen is anyone's guess. It's so absurd that there are simply no provisions about that in any of the applicable treaties and regulations.
Kicking Greece of the eurozone (which very influential people are now discussing openly) might seem like an option but it's actually very difficult to see how that would work (or differ from the current situation). Importantly, many non-EU countries (e.g. Kosovo or Montenegro) use the euro (without any control over it) and there is nothing anyone can do about that, a point made repeatedly by Dean Baker. So the ECB could further increase the pressure, e.g. by shutting down the Emergency Liquidity Assistance entirely (more on that below), or perhaps issue a strong statement but that's about all it can do to push a country out.
In any case, with no procedure in place to leave the Eurozone, Greece would retain its share of the ECB, its seat on the Governing Council, etc. even after being hung out to dry and forced to create a new currency. The only way to change that is by revising the treaties and that's only possible if all parties agree, including Greece itself.
(Incidentally, apart from this banknote-printing plot, this also implies that all the talk about a July 12 deadline or EU summit decision about a “Grexit” really refers to inviting/letting the ECB make Greek banks insolvent as detailed below because excluding Greece from the Eurozone is simply not something the European Council can vote on.)
Forcing Greece out of the European Union itself would seem even less likely. There is nothing anywhere in the treaties to allow this and it would be a huge setback for everybody, way beyond a possible “Grexit”. Besides, breaches of EU law big and small happen all the time and there is a whole range of measures European institutions can take to escalate the matter before even contemplating something as radical. One way or the other, all this would take a lot of time.
The ECB could and would however probably respond swiftly. Their main source of leverage in all this is their role in providing liquidity to the banking system and literally printing paper money cannot replace that in a modern economy. Banks have to borrow money every night from their national central bank to cover any shortage of funds (it's called “Standing Facilities”). On top of that, the ECB created something called “Emergency Liquidity Assistance” to allow national central banks to provide additional support to banks facing extraordinary pressure like in Greece at the moment.
Technically, all these mechanisms are only supposed to provide liquidity, against some collateral, to institutions that are otherwise solvent. It has been argued that the rules do not empower the ECB to use these to push for specific policies and that, if it really cared about its mandate and believed in its own “stress tests”, it would have no reason to change its mind on Greek banks. Other commentators think the ECB is already pushing the limits of what it can legally do by still supporting Greek banks in the way it did until now.
Whatever the case might be, in practice, the ECB enjoys wide discretion in choosing which banks qualify and what it accepts as collateral and it has shown itself ready to use that for political purposes. During the last week, it has maintained its support at the same level, which is not enough to fight the emerging crisis and allow the government not to impose capital controls, but enough to avoid immediate and complete collapse of the banking system. If it pulls the plug, that's what would happen and that's the main threat for Greece at the moment (not necessarily the most likely scenario even if no resolution is forthcoming, though).
One important detail is that the Governing Council takes decisions by vote (often with a two-third majority threshold I think) and has a complex votation rights rotation system. There is no need for unanimity here and no way any single member (including Greece) could block a decision. So, while the ECB Statute does not explicitly allow this, ending liquidity support is therefore one way to retaliate that could be implemented relatively easily in the current institutional framework.
But, and therein lies the rub, Greece does not want to leave the euro and is trying very very hard to avoid it. In spite of widespread perception and commentary, it has accepted unprecedented levels of austerity and many reforms, which have not been rolled back by the current government. Beyond negotiation shenanigans, Alexis Tsipras has proved ready to renege on many of his electoral promises in the name of getting a deal, without success.
Because of the mechanisms described above, printing some euro banknotes does nothing to get Greece out of this predicament. It would not be enough to keep the Greek banking system working and connected to other institutions in the Eurozone and would not help the country reach an agreement with the rest of the Eurozone.
If it were willing to give up on a negotiated solution, what Greece could do much more easily (and might still be forced to do) is issue some alternative currency-like instruments to pay pensions and public sector wages, first in the form of so-called “IOU” formally denominated in euros, which could later become a proper currency by redenominating bank deposits and accepting it as payment for taxes. But, so far, after more than one week of capital controls and a deteriorating situation with many of the downsides of an euro exit, it has resisted going down that road, which suggests that the government is not really on a “war footing” with the ECB and has absolutely no interest in something like what you propose.
Incidentally, it's not clear that the Bank of Greece would go along with a crazy idea like printing unauthorised banknotes. Like other central banks in the ECB system, it's not under the direct control of the government. Its current governor, Yannis Stournaras was appointed by the previous government, seems to remain sanguine about austerity and has reportedly had an increasingly strained relationship with the current government.
