When you make it hard to make money by trading in locale X, you will usually find that people who were making money - and paying taxes and raising GDP - in locale X - will strongly consider moving to a different locale Y, which is now more competitive with the old location as far as offering them to be domiciled there.
So, the only practical effect of such tax would be establishment of an exchange somewhere else where people can go do algorithmic trading, and they would no longer trade in your country. Instead of some increase in revenue from the tax, you get a LOSS of revenue from diminished economic activity (and thus profits to be taxed) in your country.
You already see this in hedge fund flight from NYC to CT; or with US corporatons registering in Delaware, or with assorted financial firms going to Caymans.
In case of trading and financial industry this is even worse effects wise, since they contribute very significantly to the economy of where they are (witness NYC finances - where do you think they would be if all the traders and bankers rolled up shop and moved to The City of London?)
In addition, people who moan that somehow algorithmic trading causes problems for individual investors seem to have very little understanding of capital markets.
They are claiming to be trying to solve a problem which DOES NOT EXIST. "make the market more unstable" is a nice soundbite which plays well on TV, but how does that affect individual investors?
Long term, your stock price is a function of how much the markets anticipate your company to be able to grow its profits in the future. NOTHING ELSE.
No matter how many algorithmic trades trading companies make against each other, if you as an individual investor intend to sell AAPL at 600, you will sell it at 600 (give or take - as an individual investor, the spread won't affect your profits much, and a limit trade will protect you against major volatility).
If you want to hold it till it reaches 700, no algorithmic trading or flash crashes will have any effect on long term value of your stock.
So, the only "problem" being solved here is perception that evil greedy trading companies are making "unjust" profits (the fact that said profits are more likely than not are made at the expense of competing evil greedy trading companies and not individual investors seems to be lost on the people who proposing this without having a clue about the topic they are talking about).
Oh, and there are other people who will be harmed - people whose income/wealth (you know, those evil investors like retirees) depends on how efficiently asset managers (who also use algorithmic trading) manage their retirement funds/ETFs. You don't REALLY think that they will just eat up the tax instead of passing it onto the investors in the form of lower returns, right?