This subject was raised in an article by Thom Riddle in The Ethical Spectacle of February 2004. It's worth looking deeper at the theoretical structures available for this, confirming our understanding by looking at actual historical cases then making some practical suggestions for today.
Pretty obviously "printing money" can fund a government without using taxes, sort of; when paper money was invented many governments tried it and found out the problems the hard way. It's just that if you only do that you rapidly get two problems: the money gets worthless in its own right, and you can't get people interested in your own paper - worthless a different way. Nevertheless, it has been done quite deliberately on occasion, usually in times of breakdown of more conventional approaches. In "News from Tartary" Peter Fleming records that when a 1930s Chinese warlord took over a town his priority was securing the mint, not the armoury. So printing money works, it's just that you need to do something else as well to have something workable over the long term. Keynesian economists even find it convenient to think of the printing money side of things as the main part, with taxes just a way of undoing the problems - to them, taxes aren't important at all.
Even so, printing money is a kind of tax, a tax on currency in people's possession. This insight lets us see how to make a whole range of taxes into non-taxes. That is, how to keep them from a government or an economist's point of view but rearrange them into something that people don't experience as taxes.
Here's the simplest. Corporation taxes can be rearranged into the sale of concessions. This was also a form of funding used in the 1930s Chinese warlord period. Typically it went like this. In a country with no corporations law yet, an outside company would come in with a special permission ("concession" or sometimes "capitulation"), in exchange for a fee made up partly of up front cash but mostly of shares, usually preference shares to give a priority. This rearranged a tax obligation into a dividend obligation, it didn't need special tax accounting, and it didn't need special corporations law up and running since each corporation was a one off. But it kept all the financial burdens although in a different form, and it rearranged problems of (say) transfer pricing into equivalent problems of the interests of minority shareholders.
So, where is the difference? In general, hand overs are moved up front instead of being as you go, which means they are technically fees as the government really does give something specific for them, and they are in non-cash terms so the real handovers of shares don't count as taxes. But the first part is the most important part: taxes are unrequited payments, which means the payer is always giving something for nothing, at the point of hand over. Not doing that is what makes indirect taxes less personally troubling, though they are always direct taxes on someone. In fact, during the Middle Ages nearly all government funds were handled as non-tax obligations in kind or with something being given in exchange. (Any time a "tax" is just a way of buying your way out of a non-cash obligation, like the old French "corvee" system, it isn't technically a tax even if practically everybody pays cash - it stays workable even when cash dries up in the whole economy.)
But just looking at shares, we see the government needed to manage its portfolio - technically called a "domain" - instead of relying on the tax base always being there. The company ran the sovereign risk that the government would welch, i.e. foolishly sell the shares to fund current expenditure because it looked like a good idea at the time and then turn round and say "we see you're not paying taxes, that's not fair", even though it had had the benefit when it sold out. That could even have an appearance of justice, if a former dictator had run off with the shares. At first companies didn't mind that governments could lose the lot, because neocolonialism meant that new governments didn't dare change the rules - but eventually that changed. This is pretty much what happened to the Suez Canal Company, where shares had been used to pay off an earlier Egyptian ruler but Nasser nationalised it all the same. The former ruler hadn't just blown it all either - he had tried diversifying into cotton, and lost most of it that way. That's an example of the problems of managing a domain.
The important thing to realise is that shares' value usually vanishes over time even without specially bad management or bad faith. Any portfolio needs management to stay in being since the "financial risk" grows faster than the returns - and even diversifying shares and continually rebalancing the portfolio only puts off the evil day by reducing the multiplier of the process and not the exponential factor. Of course, a diversified portfolio can include other revenue yielding assets such as land with much less financial risk - and land is how mediaeval domains mostly started off.
However if you put the two tricks together, printing money and having a diversified domain, you can apply printing fiat money to buying additional shares to make up for any erosion of the domain (the risk stays, although at a very low level, in the sense that in the end any state perishes and takes everything with it). That only leaves setting up the domain in the first place; you aren't always starting from scratch with a pool of resources you can sell off like Alaska or Alberta, or the way South Australia sold off its land to settlers at a premium and put that into revenue yielding enterprises like state railways. Even with the up front method of taking shares from every new company, people feel correctly that you are taking things from them; you could set up a starter domain by changing the existing tax commitments for a slice of shares, a "composition" of the existing tax obligation, but if you went on asking for new shares from new companies people would start getting upset at that.
So it works better if governments simply buy shares from all new companies, printing money to do it; they then need a "sinking fund" approach to set aside some dividends and undo inflation, a function very like the original purpose of sinking funds, setting up matching investments to cover commitments to pay out on national debts. Over time, no net new money gets printed so it only has an effect on rebalancing the domain and keeping it in being. This is a counsel of perfection; all governments ignore the other things they need to do. Sinking funds never did work because governments just went into debt again, and whenever there are domains governments try to sell them off or claim that their own pet projects are "really" investment and that it's quite proper to sell off existing investments to put funds into those. We can see this quite clearly by contrasting what happened with the Alaska Permanent Fund and Alberta Heritage Fund - the Alaskan one was entrenched and constitutionally protected to keep the politicians off, but they had a free rein in Alberta.
The thing is, with money just being printed but being paid out that way, it creates real productive capacity as well as its usual downside - and the sinking fund approach uses the gain to unwind the harm gradually. This is not what happens with printing money just to spend and not what happens with taking shares as an up front obligation without the government really doing anything except register the company. With the Suez Canal Company, the Khedive made local labour available to dig the canal under the direction of European engineers, and with Alaska and Alberta there is a hand over of real mineral resources.
What is the equivalent when a government just registers a company? It's setting up something to compete with real people and partnerships, which are much more natural entities. So, indirectly and even though it's not technically a tax, it's having the economic effect of taxing all of us by giving a commercial advantage to the companies. It's better if we get taxed indirectly with printed money, since at least that way real investments get made (diverting our consumption to do it, by making it more expensive with inflation); the real investment eventually benefits us with trickle down.
Oh - printing money to buy shares from existing shareholders in existing corporations isn't productive, it's just doing a wealth transfer (in fact, US investment overseas is partly of this sort, ripping off the other countries by exporting US inflation to acquire existing overseas resources for nothing but depreciating paper).
Even so, big government makes big distortions to the economy this way; it only works as much as it does in Alaska and Alberta as those domains aren't making their economies top heavy. It's really important that governments are always quite small minority shareholders, as otherwise the tail wags the dog and people can't exercise their self interest and make proper business decisions - the Alberta temptation makes governments do things other than for revenue. From that point of view, today's governments are at least twice as big as they should be to avoid distorting the economies that are carrying them.
So it's best to reduce the size of governments and do more direct investment. This can be done, the way the Dutch organised the East Indies into the "culture system" to get cash crops or the way the British got rent from Indian land organised under the Ryot tenure system, but those really worked with the help of middlemen (the Dutch even used that "printing money" trick, setting up their system by depreciating the coinage to pay for it).
That shows that government investment is best handled going through smaller units like local government or the middlemen in India or the Dutch East Indies. In former times, universities were given endowments of land so that those resources would do the work - governments stayed clear of the education sector. Even individuals could benefit, like the famous Danish astronomer Tycho Brahe who was given the use of an entire island to help with his researches. Many US universities had something similar, subsidised by land releases and even using a scrip money system connected with it, the same connection of money and a domain I outlined just above (Senator Morrill was associated with this scheme, as well as with many other financial things of the day). Of course, today's governments want to keep the strings attached and the public services under their control, but history shows that this can work, at least up to a point.
It seems to me that the most practical way to handle this approach in our age is, with municipalities handling most of the direct resources, renting out their own urban land rather than having land taxes and applying new funds to fleet leasing of vehicles and other equipment for the help of local businesses (including rural enterprises, which is the way to get revenue from rural areas). That would keep the business risk where it belongs, with business owners and operators who would have an enlightened interest in trying to do the right things, but the municipalities would have a larger revenue base than just with land taxes. With this approach, the central government would get its slice from municipal bonds, which it would buy with printed money whenever a municipality did an issue for any genuine revenue yielding investment (only taking a proportion to keep the price set by individual investors, of course).
However it is likely that customs and excises - taxes on goods and services in and out of the country, and high value or specialised goods and services - would still be needed. This is partly because it would be difficult to get governments down to a size affordable this way, and partly because it would be necessary to compartment off an economy like this from economies running in different ways - a sort of quarantine. Nevertheless, there is no need for taxes paid by natural persons in their own right, apart from a problem of transition.
The author, P.M.Lawrence, has some other publications here, many on economic issues.