Would this work?
Barely, badly, and for some time.
(and it would depend on what exactly you call "printing money".)
How long, how badly, and how barely, it would depend on the situation of your economy, previous to such inovation.
The magical/sacred equation of monetary policy - the one economists repeat as a mantra, their everynight prayer to Mammon - is this small piece of elegance:
MV ≅ PQ
in which M is the amount of circulating money, V is the speed (velocity) of money circulation, P is the general level of prices, and Q is the total value of everything produced in a given economy.
Your hypothesis is basically increasing M:
MV ≅ PQ ---> MV ≅ PQ
Problem is, if you increase M, then one of these three things must happen: V will decrease, or either P or Q will increase. (Evidently, it could happen that V will decrease a little, and P and Q will increase a little. But let's go with the "pure" cases).
First, perhaps V decreases:
MV ≅ PQ ---> Mv ≅ PQ
This first possibility is not well studied; economists usually assume that V is inelastic, or that it will only change due to specific changes on the technology of circulation (such as the introduction of paper money or virtual money - and these changes would usually go in the way of increasing V).
So, in practice, economists will offer a few human sacrifices to Mammon in order to get
MV ≅ PQ ---> MV ≅ PQ
in which the increase of circulating money translates into an increase of production. In fact, with very few exceptions, governments do exactly this to a point, with much caution, in order to get their economies to grow (because a static M will make growth very difficult).
But Mammon being a capricious god, what you are more likely to get is,
MV ≅ PQ ---> MV ≅ PQ
in which, instead of production growing, prices rise.
This will allow us to answer the questions, "how long, how badly, and how barely" would this work, as long as we understand how Mammon decides between the increase in P versus the increase in Q.
Prices are relatively easy to increase, especially if the economy is considerably oligopolistic, and especially in the case of products whose demand is inelastic (if the prices of, say, videogames, rise, people can buy less videogames - the demand is elastic - but if prices of food rise, people will have to pay more or starve - the demand is inelastic). Production is much more difficult to increase, because it will demand more investments. Surplus money is one condition for investment, but there are others. And investments' returns may be delayed in time; even if entrepreneurs decide to invest, thus pointing to an increase in Q, it may be that the increase in Q is only going to happen once their new factories or new machines are functioning, which may be several months to a few years into the future. Which means that P will tend to rise in the short term, even if there is investment.
So, we can see that an increase in M will more likely result in an increase in P if:
- The economy is highly oligopolic;
- The economy relies heavily in inelastic goods - generally, primary products of immediate consumption;
- The economy is already functioning at or near at the limits of its capacity.
Conversely, an increase in M will more likely result in an increase in Q if:
- The degree of monopolisation of the economy is low;
- The economy relies heavily in elastic goods;
- The economy is functioning far from the limits of its capacity.
So, a heavily industrialised but not very heavily oligopolic economy that is just coming out of a recession will probably respond quite well to your scheme, while an oligopolic economy heavily based in agriculture that is already overbusy will probably go into inflation - high inflation - hyperinflation - political crisis very quickly.
Then the problem is that, even in the first case, the growth of the economy caused by the expansion of M itself will eventually make it come closer to its capacity. As long as production can be increased by simply hiring more workers for longer hours and increasing the intensity of the use of machines - additional shifts, for instance - things will go relatively well. But at some point, further increases will demand more or newer machines or bigger factories, and this will only result in an increase in Q in a longer term than what is needed for things to go smoothly. At this point, inflation is going to be unavoidable.
And then the government will want to reintroduce taxes, but this is going to be much more difficult in an economy already facing serious problems of inflation.
And then, it depends of what you call "printing money".
Because actually printing bills, or coining coins, is only a small part of monetary policy, and bills and coins are a quite small part of actual circulating money, so expanding them will probably have little effect in the economy as a whole (see, for instance, definition of M0, M1, etc and money supply in India). When a government really wants to expand M (which, at least in part, means it wants to fund itself through monetary policy) it expands credit. Central banks lend money to commercial banks, and allow those banks to re-lend money to other economic agents (generally, firms and families), but it imposes a reserve requirement that limits the creation of new virtual money by banks. When a government wants to expand its circulating money, it reduces the reserve requirement, which in turn makes credit easier and cheaper, which has either the intended result of economic growth, or the much feared result of inflation.
This brings an additional problem: since any abrupt increase in reserve requirements will probably precipitate an immediate monetary crisis and a recession, governments tend to try to reduce the money supply by borrowing money back, thus reducing the amount of circulation money. But this can cause the government to increase interests too much, and get caught in a vicious cycle: borrowing more money at higher interest rates to decrease M, and then having to increase M in order to remain potentially able to pay back its debts (the kind of conundrum that used to recurrently plague Brasil up to the end of the 20th century, for instance).
How would this affect the faith in the currency system?
Potentially, it could destroy such faith, through hyperinflation.
Which class would be relatively taxed the most?
All those who rely on relatively fixed prices: workers first and foremost, landlords, especially urban owners of for-rent aparments; also importers (who will find the foreign currency the need to import goods ever more expensive).
In short, it is not exactly a good idea; taxes are more reliable, have less negative effects, and are more easier to scale down (and politically much more difficult to raise if necessary).